Showing papers on "Conformal map published in 1990"

Conformal Invariance and Pion Wave Functions of Nonleading Twist

Abstract: The restrictions are studied for the general structure of pion wave functions of twist 3 and twist 4 imposed by the conformal symmetry and the equations of motion. A systematic expansion of wave functions in the conformal spin is built and the first order corrections to asymptotic formulae are calculated by the QCD sum rule method. In particular, we have found a multiplicatively renormalizable contribution into the two-particle wave function of twist 4 which cannot be expanded in a finite set of Gegenbauer polynonials.

Plane Stokes flow driven by capillarity on a free surface

Abstract: The free creeping viscous incompressible plane flow of a finite region, bounded by a simple smooth closed curve and driven solely by surface tension, is analysed. The shape evolution is described in terms of a time-dependent mapping function z = Ω (ζ, t) of the unit circle, conformal on |ζ| [les ] 1. An equation giving the time evolution of the Ω (ζ, t) is derived. In practice, it has been necessary to guess a parametric form, i.e. Ω (ζ, t) = Ω[ζ; a1(t), a2(t), …], whose validity must be verified using the shape-evolution equation. Polynomial and proper rational mappings with no repeated factors are apparently always valid in principle. Solutions are given for (i) regions bounded initially by a regular epitrochoid, (ii) the limiting case of a half-plane bounded by a trochoid, and (iii) a class of rosettes whose mapping is rational. The two-lobed rosette gives the exact solution of the coalescence of equal cylinders. All these mappings involve limiting initial shapes having inward-pointing cusps. Useful parameterizations providing regions whose limiting shapes possess corners or outward-pointing cusps have not been found.

Design considerations for a space-variant visual sensor with complex-logarithmic geometry

Abstract: A space-variant sensor design based on the conformal mapping of the half disk, w=log (z+a), with real a>0, which characterizes the anatomical structure of the primate and human visual systems is discussed. There are three relevant parameters: the circumferential index kappa which is defined as the number of pixels around the periphery of the sensor, the visual field radius R (of the half-disk to be mapped), and the map parameter a, which displaces the logarithm's singularity at the origin out of the domain of the mapping. It is shown that the log sensor requires O( kappa /sup 2/log (R/a)) pixels. An analysis is presented which makes it possible to compare directly the space complexity of different sensor designs in the complex logarithmic family. In particular, rough estimates can be obtained of the parameters necessary to duplicate the field width/resolution performance of the human visual system. >

Genus-one path integral in two-dimensional quantum gravity.

Abstract: We calculate the path integral on a torus in continuum Liouville theory coupled to conformal matter with c\ensuremath{\le}1. We find agreement with the results obtained in matrix models, topological gravity, and Korteweg\char21{}de Vries hierarchies.

A modified Schwarz-Christoffel transformation for elongated regions

Abstract: The numerical computation of a conformal map from a disk or a half plane onto an elongated region is frequently difficult, or impossible, because of the so-called crowding phenomenon. This paper shows that this problem can often be avoided by using another elongated region, an infinite strip, as the standard domain. A transformation similar to the Schwarz–Christoffel formula maps this strip onto an arbitrary polygonal channel, and a slightly modified transformation maps an elongated rectangle onto an arbitrary closed polygon. By using robust and efficient software for numerical integration and solution of the parameter problem, high-accuracy maps of distorted regions with aspect ratios as high as thousands to one are constructed. The modified mapping method has natural applications in fluid mechanics and electrical engineering.

Conformal generalized near‐field acoustic holography for axisymmetric geometries

Abstract: The normal velocity field on the boundary σ of a radiating source is reconstructed with high resolution by using pressure data, measured in the very near field to include evanescent components, fast decaying with the distance. The measurement points belong to a surface conformal to σ. The reconstruction is performed by identifying the boundary normal velocity distribution generating the pressure field best mean‐square fitting the measurement data. The ensuing normal equations are solved by resorting to the singular value decomposition (SVD) of the transformation from the boundary normal velocity to the pressure at the measurement points. The reconstructed boundary normal velocity is represented as a linear combination of basis functions, each associated with a singular value (SV). The errors in the estimation of the coefficients of the linear combination are proportional to the inverses of the corresponding SVs. To achieve robustness, the SVD is truncated to retain the terms corresponding to a limited dynamic range, in turn related to the dynamic range of the measurement system. For axisymmetric, but otherwise general, surfaces, the numerical complexity is reduced because separate SVDs of greatly reduced dimensionality must be performed on a number of matrices, each associated with one of the harmonics of the Fourier expansion of the field around the symmetry axis. In a numerical experiment, making use of an available computer model, by processing very‐near‐field pressure samples, the normal velocity was reconstructed with excellent accuracy on the surface of a finite compliant cylinder with flat endcaps, excited by point forces. The viability of the method was then confirmed in a measurement tank experiment in which the distribution of the boundary normal velocity was reconstructed on a finite compliant cylinder driven by point shakers. Comparison with accelerometer data shows good agreement on a wide frequency band.

Conformal symmetries and fixed points in space‐time

Abstract: A general study is made of conformal vector fields on a four‐dimensional Lorentz manifold with particular emphasis being laid on the structure of the zeros (critical points) of such vectors fields. The implications for general relativity are investigated and a discussion of conformal vector fields in generalized plane wave space‐times is given. An attempt is made to clarify the well‐known theorem of Bilyalov and Defrise‐Carter.

On the geometry of harmonic morphisms

Abstract: Let π:M→B be a horizontally conformal submersion. We give necessary curvature conditions on the manifolds M and B, which lead to non-existence results for certain horizontally conformal maps, and harmonic morphisms. We then classify all such maps between open subsets of Euclidean spaces, which additionally have totally geodesic fibres and are horizontally homothetic. They are orthogonal projections on each connected component, followed by a homothety.

Quasiextremal distance domains and conformal mappings onto circle domains

Abstract: This paper contributes to the theory of quasiextremal distance domains. We present some new properties for these domains and point out results concerning the extension of quasiconformal homeomorphisms.For example, we establish a continuity property for mod(E, F; D) and use this to demonstrate that mod(E, F; D) = cap(E, F; D) whenever D is a QED domain and E, F are disjoint compacta in D. Our final result is that if each boundary component of a plane domain is either a point or a Jordan curve and if the domain satisfies a boundary quasiextremal distance property, then there exists a quasiconformal self-homeomorphism of the entire plane which maps the given domain conformally onto a circle domain.

Circle packings in the approximation of conformal mappings

Abstract: Connections between circle packings and analytic functions were first suggested by William Thurston [T2], who conjectured that the conformai mapping of a simply connected plane domain Q to the unit disc A could be approximated by manipulating hexagonal circle configurations lying in Q. The conjecture was confirmed by Rodin and Sullivan [RS]. Their proof relies heavily on the hexagonal combinatorics of the circle configurations, a restriction not suggested by the underlying intuition. The purpose of this note is to announce that Thurston's conjecture is true with much weaker combinatoric hypotheses and to outline the proof. The main lines of argument are those developed by Rodin and Sullivan, but the proof is independent. The deepest part of their work—a uniqueness result of Sullivan's which depends on Mostow rigidity—is replaced here by probabilistic arguments. We work in the setting of hyperbolic geometry and make use of the discrete Schwarz-Pick lemma proven in [BS] to understand the behavior of circle configurations. We analyze how curvature distributes itself around a packing as successive differential changes are made to boundary circles, ultimately modelling this process as a random walk. The proof that a certain limiting random walk is recurrent replaces the uniqueness result of Sullivan. Details will appear elsewhere. The author gratefully acknowledges support of the National Science Foundation and the Tennessee Science Alliance.

Conformal circles and parametrizations of curves in conformal manifolds

Abstract: We give a simple ODE for the conformal circles on a conformal manifold, which gives the curves together with a family of preferred parametrizations. These parametrizations endow each conformal circle with a projective structure. The equation splits into two pieces, one of which gives the conformal circles independent of any parameterization, and another which can be applied to any curve to generate explicitly the projective structure which it inherits from the ambient conformal structure [1]. We discuss briefly the use of conformal circles to give preferred coordinates and metrics in the neighborhood of a point, and sketch the relationship with twistor theory in the case of dimension four.

Biofluiddynamics of balistiform and gymnotiform locomotion. Part 2. The pressure distribution arising in two-dimensional irrotational flow from a general symmetrical motion of a flexible flat plate normal to itself

Abstract: When, for an otherwise unbounded fluid, the unique irrotational flow compatible with the instantaneous motion of an immersed body has been calculated, it is straightforward to deduce the pressure field from the unsteady form of Bernoulli's equation if the body is rigid. On the other hand, if the body is flexible, a somewhat subtle analysis is required to determine the time derivative of velocity potential, ∂ϕ/∂t, which occurs in that equation. This is because no simple relationship exists between the instantaneous form of ϕ and its form at a nearby instant.In the case of two-dimensional flow, however, the two forms of ϕ for a flexible body may be related, not in general by a simple translational and/or rotational mapping as for a rigid-body motion, but by a conformal mapping. The example of a flexible flat plate is used here to illustrate this approach to calculating the pressure field.In the analysis of balistiform motion by elongated-body theory (Lighthill & Blake 1990), one part of the propulsive force on the fish has magnitude equal to P, the area integral of the pressure field just described. This area integral is shown in §3 below to take a simple form or for E, so that P itself is also not enhanced. For the relevance of these findings to the efficiency of balistiform motion, see Lighthill & Blake (1990).

Conformal theories, curved phase spaces, relativistic wavelets and the geometry of complex domains

Abstract: We investigate some aspects of complex geometry in relation with possible applications to quantization, relativistic phase spaces, conformal field theories, general relativity and the music of two and three-dimensional spheres.

Potential Flow in Channels

Abstract: A method is presented for calculating potential flows in infinite channels. Given a collection of N sources in the channel and a zero normal flow boundary condition, the method requires an amount of work proportional to N to evaluate the induced velocity field at each source position. It is accurate to within machine precision and for its performance does not depend on the distribution of the sources. Like the Fast Multipole Method developed by Greengard and Rokhlin [J. Comput. Phys., 73 (1987), pp. 325–348], it is based on a recursive subdivision of space, knowledge of the governing Green's function, and the use of asymptotic representations of the potential field. Previous schemes have been based either on conformal mapping, which experiences numerical difficulties with the domain boundary, or direct evaluation of Green's function. Both require $O(N^2 )$ work.

The Schwarzian derivative for conformal maps.

Abstract: Let / be an analytic map and z a (non-critical) point of the domain. O. Martio and J. Sarvas [6] showed that there is a M bius transformation T which approximates / at z in the sense that it agrees with / up to the second derivative at z. The difference in the third derivatives is determined by the Schwarzian derivative Sf (z) of / at z. (See also W. Thurston [13].) The purpose of this paper is t o prove the corresponding result for conformal mappings /: M —* M between Riemannian manifolds of any dimension. For each χ e M there is a unique M bius transformation T from the tangent space to M at χ onto the tangent space to M at fx which agrees with / up to the second derivative. The deviation of / from T is measured by the \"Schwarzian derivative\" I f x which is a Symmetrie linear map from the tangent space at χ to itself.

Beltrami differentials, conformal models and their supersymmetric generalizations

Abstract: Conformally invariant couplings of two-dimensional field theories to gravity can be formulated either in a Riemannian surface or manifold (metric) framework After a detailed presentation of the Riemannian surface approach (and comparison with the metric formalism), the authors develop its supersymmetric generalization Super Beltrami differentials are introduced without any reference to metric (vielbein) structures and superconformal models are studied in this framework The main goal of the work consists in the construction of local (and supersymmetric) field theories defined on arbitrary (super-)Riemann surfaces and exhibiting the (super-)holomorphic factorization in a manifest way

Full-wave analysis of coplanar waveguides by variational conformal mapping technique

Abstract: A new full-wave analysis of coplanar waveguides is presented. The modified mapping technique of C.P. Wen (1969) is used to map the original infinite domain into a finite image domain and also to account for the singularity of fields near the conductor edges. The finite thickness of the dielectric substrate is considered together with the assumptions of lossless guides and negligible metallization thickness. The current distributions on the center signal strip as well as the tangential electric fields over the slot along the air-dielectric interface are examined. Numerical results for the frequency-dependent effective dielectric constants and characteristic impedances of coplanar waveguides are presented. Particular attention is given to the electric field distributions over the air-dielectric interface of slots and the current distributions of the signal strip. >

Computation of conformal maps by modified schwarz-christoffel transformations

Abstract: In this thesis we explore four issues regarding the use of the Schwarz-Christoffel transformation and related methods as robust conformal mapping techniques. The first of these, the crowding phenomenon, makes maps onto even mildly elongated regions so ill-conditioned they may be impossible to calculate. We overcome this difficulty by implementing a transformation, similar to the Schwarz-Christoffel formula, which uses an infinite strip as the standard domain. The mapping function itself is not original; it has previously been used by others to compute maps onto open channels for internal flow computations. The interpretation in terms of crowding is original, however, and we also present a new modification for mapping elongated rectangles onto closed polygons. By using robust and efficient software for numerical integration and solution of the parameter problem, we successfully map difficult regions with aspect ratios of up to thousands to one. Second, we survey a number of techniques for numerically evaluating Schwarz-Christoffel integrals, including the compound Gauss-Jacobi algorithm, singularity removal methods, and several adaptive quadrature subroutines. The Gauss-Jacobi method is shown to be faster than its nearest competitor by roughly a factor of two; we recommend it for all applications unless strict error bounds are required. Third, all Schwarz-Christoffel methods require the solution of a system of nonlinear algebraic equations. We examine two approaches to this problem that have appeared in the literature, Davis's iterative scheme and Trefethen's formulation as a nonlinear system. We show that Davis's method is difficult to generalize and is not always even locally convergent, and prove that although Trefethen's method never diverges locally, it too may diverge from distant starting points. A generalization of the Schwarz-Christoffel formula describes conformal maps onto regions bounded by arbitrary arcs of circles as solutions to a nonlinear ordinary differential equation. We present improvements in methods for integration of the differential equation, normalization of approximate solutions, specification of the accessory parameters, formulation of the nonlinear system, and selection of appropriate starting points involving a minimum amount of distortion. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) (Abstract shortened with permission of school.)

Special conformal symmetries in general relativity

Abstract: A geometrical discussion of special conformal vector fields in space-time is given. In particular, it is shown that if such a vector fieldξ is admitted, it is unique up to a constant scaling and the addition of a homothetic or a Killing vector field. In the case when the gradient of the conformal scalar associated withξ is non-null it is shown that other homothetic and affine symmetries are necessarily admitted by the space-time, that an intrinsic family of 2-dimensional flat submanifolds is determined in the space-time, thatξ is, in general, hypersurface orthogonal and that the space-time, if non-flat, is necessarily (geodesically) incomplete. Other geometrical features of such space-times are also considered.

Semi-infinite two-dimensional Ising model with marginally inhomogeneous couplings and conformal invariance

Abstract: The authors consider a semi-infinite two-dimensional Ising model with nearest-neighbour couplings that deviate from the bulk critical couplings by Al-2, where l is the distance from the surface The surface critical exponents of this system are non-universal Under a conformal mapping onto a strip of width L, the Al-1 inhomogeneity transforms into A((L/ pi )sin( pi L/L))-1 For the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit The analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance

Exact solutions of semi-infinite crack in a strip

Abstract: This study presents a procedure for solving semi-infinite crack which is a very useful model in geophysics, seismology and engineering. With the help of conformal transformation mapping the region in a physical plane onto the interior of a unit circle in ζ plane, any case of this kind of problems can be solved unifiedly, and some exact solutions are obtained

Sewing Polyakov amplitudes. I. Sewing at a fixed conformal structure

Abstract: We consider the problem of reconstructing the correlation functions of a conformal field theory on a surface Σ from the correlation functions on a surface Σ′ obtained from Σ by cutting along a closed curve. We show that under quite general conditions, the correlation functions on the cut surface can be “sewn” by integrating over appropriate boundary valuess of the field.

Asymptotic mass degeneracies in conformal field theories

Abstract: By applying a method of Hardy and Ramanujan to characters of rational conformal field theories, we find an asymptotic expansion for degeneracy of states in the limit of large mass which isexact for strings propagating in more than two uncompactified space-time dimensions. Moreover we explore how the rationality of the conformal theory is reflected in the degeneracy of states. We also consider the one loop partition function for strings, restricted to physical states, for arbitrary (irrational) conformal theories, and obtain an asymptotic expansion for it in the limit that the torus degenerates. This expansion depends only on the spectrum of (physical and unphysical) relevant operators in the theory. We see how rationality is consistent with the smoothness of mass degeneracies as a function of moduli.

Braiding matrices, modular transformations and topological field theories in 2+1 dimensions

Abstract: Relations between 3D topological field theories and rational conformal field theories are discussed. In the former framework, we can define the generalized Verlinde operators. Using these operators, we find modular transformations for conformal blocks of one point functions and two point functions on the torus. The result is generalized to higher genus. The correctness of our formulae is illustrated by some examples. We also emphasize the importance of the fusion algebra.