Showing papers on "Singularity published in 1992"
458 citations
01 Jan 1992
315 citations
TL;DR: In this article, a relation between the spectrum of solitons of massive quantum field theories and the scaling dimensions of chiral fields at the conformal point has been found, where the condition that scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetric $N=2$ conformal theories and their massive deformations.
Abstract: We find a relation between the spectrum of solitons of massive $N=2$ quantum field theories in $d=2$ and the scaling dimensions of chiral fields at the conformal point. The condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetric $N=2$ conformal theories and their massive deformations in terms of a suitable generalization of Dynkin diagrams (which coincides with the A--D--E Dynkin diagrams for minimal models). The Landau-Ginzburg theories are a proper subset of this classification. In the particular case of LG theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity; the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of Picard-Lefschetz theory.
280 citations
01 Jan 1992
TL;DR: In this paper, the authors studied the properties of tangent vectors and tangent planes in the neighborhood of singular points in analytic manifolds. But their local geometric properties are less well understood.
Abstract: Algebraic and analytic varieties have become increasingly important in recent years, both in the complex and the real case. Their local structure has been intensively investigated, by algebraic and by analytic means. Local geometric properties are less well understood. Our principal purpose here is to study properties of tangent vectors and tangent planes in the neighborhood of singular points. We study stratifications of an analytic variety into analytic manifolds; in particular, we may require that a transversal to a stratum is also tranversal to the higher dimensional strata near a given point. A conjecture on possible fiberings of the variety (and of surrounding space) is stated; it is proved at points of strata of codimension 2 in the surrounding space. In the last sections, we see that a variety may have numbers or functions intrinsically attached at points or along strata; also an analytic variety may be locally unlike an algebraic variety.
255 citations
TL;DR: The formation and semi-classical evaporation of two-dimensional black holes is studied in an exactly solvable model where infalling matter is reflected from a time-like naked singularity and all information is recovered at spatial infinity.
Abstract: The formation and semi-classical evaporation of two-dimensional black holes is studied in an exactly solvable model. Above a certain threshold energy flux, collapsing matter forms a singularity inside an apparent horizon. As the black hole evaporates the apparent horizon recedes and meets the singularity in a finite proper time. The singularity emerges naked and future evolution of the geometry requires boundary conditions to be imposed there. There is a natural choice of boundary conditions which match the evaporated black hole solution onto the linear dilaton vacuum. Below the threshold energy flux no horizon forms and boundary conditions can be imposed where infalling matter is reflected from a time-like naked singularity. All information is recovered at spatial infinity in this case.
217 citations
TL;DR: In this article, the formation and semi-classical evaporation of two-dimensional black holes is studied in an exactly solvable model and boundary conditions are imposed where infalling matter is reflected from a time-like naked singularity.
Abstract: The formation and semi-classical evaporation of two-dimensional black holes is studied in an exactly solvable model. Above a certain threshold energy flux, collapsing matter forms a singularity inside an apparent horizon. As the black hole evaporates the apparent horizon recedes and meets the singularity in a finite proper time. The singularity emerges naked and future evolution of the geometry requires boundary conditions to be imposed there. There is a natural choice of boundary conditions which match the evaporated black hole solution onto the linear dilaton vacuum. Below the threshold energy flux no horizon forms and boundary conditions can be imposed where infalling matter is reflected from a time-like naked singularity. All information is recovered at spatial infinity in this case.
198 citations
TL;DR: In this paper, a unified account of the Schwarz function and conformal mapping approaches for the zero-surface-tension case is given, and a new interpretation of the linear stability analysis of the zero surface tension problem is proposed, and the authors suggest a regularization of ill-posed problems by the imposition of a unilateral constraint on the moving boundary.
Abstract: We discuss the one-phase Hele–Shaw problem in two space dimensions. We review exact solutions in the zero-surface-tension case, giving a unified account of the Schwarz function and conformal mapping approaches. We discuss the extension of the former method to the cases in which surface tension or ‘kinetic undercooling’ terms apply on the moving boundary, and we give some conjectures on the resulting singularity structure. Finally, we give a new interpretation of the linear stability analysis of the zero-surface-tension problem, and we suggest a possible regularization of ill-posed problems by the imposition of a unilateral constraint on the moving boundary.
194 citations
TL;DR: In this article, the integrability aspects of a classical one-dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter "a" are studied.
Abstract: The integrability aspects of a classical one‐dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter ‘‘a’’ are studied. Through a differential geometric approach, the dynamical equation for the spin chain is expressed in the form of a higher‐order generalized nonlinear Schrodinger equation (GNLSE). An integrable biquadratic chain that is a deformation of the lower‐order continuum isotropic spin chain, is identified by carrying out a Painleve singularity structure analysis on the GNLSE (also through gauge analysis) and its properties are discussed briefly. For the nonintegrable chain, the perturbed soliton solution is obtained through a multiple scale analysis.
193 citations
TL;DR: The formation and quantum mechanical evaporation of black holes in two spacetime dimensions can be studied using effective classical field equations, recently introduced by Callan et al. as mentioned in this paper.
163 citations
CERN1
TL;DR: In this paper, the existence of a non-degenerate quasihomogeneous polynomial in a configuration has been shown to be possible in the space of polynomials with a fixed set of weights.
Abstract: We give a criterion for the existence of a non-degenerate quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition derived from the formula for the Poincare polynomial. We further prove finiteness of the number of configurations for a given value of the singularity index. For the value 3 of this index, which is of particular interest in string theory, a constructive version of this proof implies an algorithm for the calculation of all non-degenerate configurations.
162 citations
TL;DR: In this article, the motion of periodic vortex sheets is computed using a new, spectrally accurate approximation to the Birkhoff-Rott integral, where the catastrophic effect of roundoff error is suppressed by application of a Fourier filter, which itself operates near the level of the round-off.
Abstract: Moore's asymptotic analysis of vortex-sheet motion predicts that the Kelvin–Helmholtz instability leads to the formation of a weak singularity in the sheet profile at a finite time. The numerical studies of Meiron. Baker & Orszag, and of Krasny, provide only a partial validation of his analysis. In this work, the motion of periodic vortex sheets is computed using a new, spectrally accurate approximation to the Birkhoff–Rott integral. As advocated by Krasny, the catastrophic effect of round-off error is suppressed by application of a Fourier filter, which itself operates near the level of the round-off. It is found that to capture the correct asymptotic behaviour of the spectrum, the calculations must be performed in very high precision, and second-order terms must be included in the Ansatz to the spectrum. The numerical calculations proceed from the initial conditions first considered by Meiron, Baker & Orszag. For the range of amplitudes considered here, the results indicate that Moore's analysis is valid only at times well before the singularity time. Near the singularity time the form of the singularity departs away from that predicted by Moore, with the real and imaginary parts of the solutions becoming differentiated in their behaviour; the real part behaves in accordance with Moore's prediction, while the singularity in the imaginary part weakens. In addition, the form of the singularity apparently depends upon the initial amplitude of the disturbance, with the results suggesting that either Moore's analysis gives the complete form of the singularity only in the zero amplitude limit, or that the initial data considered here is not yet sufficiently small for the behaviour to be properly described by the asymptotic analysis. Convergence of the numerical solution beyond the singularity time is not observed.
TL;DR: In this paper, a family of product integration rules for weakly singular and hypersingular integrals is presented, which are hierarchically constructed from "finite part" integration formulas in radial and Gaussian formulas in angular direction.
Abstract: We present and analyze methods for the accurate and efficient evaluation of weakly, Cauchy and hypersingular integrals over piecewise analytic curved surfaces in ?3.
The class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three-dimensional domains by the boundary integral equation method. The possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point. This rules out weighted quadrature approaches with a fixed singular weight.
For weakly singular integrals it is shown that Duffy's triangular coordinates leadalways to a removal of the kernel singularity. Also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero. These are based on the Rabinowitz-Richter estimates in connection with an asymptotic estimate of domains of analyticity in ?2.
It is further shown that the modified extrapolation approach due to Lyness is in the weakly singular case always applicable. Corresponding error and asymptotic work estimates are presented.
For the weakly singular as well as for Cauchy and hypersingular integrals which e.g. arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates. These rules are hierarchically constructed from "finite part" integration formulas in radial and Gaussian formulas in angular direction. Again, we show how the Rabinowitz-Richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size.
TL;DR: Results for the behavior of vortex density across the transition and how the dilute-gas approximation breaks down are presented and the exponent {eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods.
Abstract: We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat ${\mathit{C}}_{\mathit{v}}$ for the XY model. The simulations are done on ${64}^{2}$, ${128}^{2}$, ${256}^{2}$, and ${512}^{2}$ lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{
u}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, ${\mathit{T}}_{\mathit{c}}$=0.894(5), is obtained using KT fits with \ensuremath{
u} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.
TL;DR: In this paper, it was shown that for any given points x 1,…,xk in I° there is a solution u such that its blow-up points are exactly x1, […], xk.
Abstract: We consider the equation
where I ⊂ ℝ u is scalar-valued and p > 1.
It has been proven that if u(t) blows up at time T, the blow-up points are finite in number and located in I°.
Our aim is to prove that this result is optimal. That is, for any given points x1,…,xk in I° there is a solution u such that its blow-up points are exactly x1,…,xk.
TL;DR: In this article, the authors developed a thermodynamic formalism for a large class of maps of the interval with indifferent fixed points, which yields onedimensional systems with many-body infinite-range interactions for which the thermodynamics is well defined but Gibbs states are not.
Abstract: We develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points. For such systems the formalism yields onedimensional systems with many-body infinite-range interactions for which the thermodynamics is well defined but Gibbs states are not. (Piecewise linear systems of this kind yield the soluble, in a sense, Fisher models.) We prove that such systems exhibit phase transitions, the order of which depends on the behavior at the indifferent fixed points. We obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures. Our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable (expanding) induced version. The technique allows one also to obtain a version of the Bowen-Ruelle formula for the Hausdorff dimension of repellers for maps with indifferent fixed points, and to generalize Fisher results to some nonsoluble models.
TL;DR: In this paper, a family of p-method plane elasticity elements is derived based on the hybrid Trefftz formulation, where exact solutions of the Lame-Navier equations are used for the intra-element displacement field together with an independent displacement frame function field along the element boundary.
Abstract: A family of p-method plane elasticity elements is derived based on the hybrid Trefftz formulation.1 Exact solutions of the Lame-Navier equations are used for the intra-element displacement field together with an independent displacement frame function field along the element boundary. The final unknowns are the parameters of the frame function field consisting of the usual degrees of freedom at corner nodes and an optional number of hierarchic degrees of freedom associated with the mid-side nodes. Since the element matrices do not involve integration over the element area, the elements have a polygonal contour with an optional number of curved sides. The quadrilateral element has the same external appearance as the conventional p-method plane elasticity element.2,3 But unlike in the conventional p-method approach, suitable special-purpose Trefftz functions are generally used to handle the singularity and/or stress concentration problems rather than a local mesh refinement. The practical efficiency of the new elements is assessed through a series of examples.
TL;DR: Results indicate that there is a naked singularity in black-hole evaporation, or (more likely) that the semiclassical approximation breaks down.
Abstract: An interesting two-dimensional model theory has been proposed that allows one to consider black-hole evaporation in the semiclassical approximation. The semiclassical equations will give a singularity where the dilaton field reaches a certain critical value. This singularity will be hidden behind a horizon. As the evaporation proceeds, the dilaton field on the horizon will approach the critical value but the temperature and rate of emission will remain finite. These results indicate either that there is a naked singularity, or (more likely) that the semiclassical approximation breaks down.
TL;DR: It is shown that there can be no simple relationship between distance to singularity and condition number, because the condition number can be computed in pol...
Abstract: The singular value decomposition of a square matrix A answers two questions First, it measures the distance from A to the nearest singular matrix, measuring distance with the two-norm It also computes the condition number, or the sensitivity of $A^{ - 1} $ to perturbations in A, where sensitivity is also measured with the two-norm As is well known, these two quantities, the minimum distance to singularity and the condition number, are essentially reciprocals Using the algorithm of Golub and Kahan [SIAM J Numer Anal, Ser B, 2 (1965), pp 205–224] and its descendants, these quantities may be computed in $O(n^3 )$ operations More recent sensitivity analysis extends this analysis to perturbations of different maximum sizes in each entry of A One may again ask about distance to singularity, condition numbers, and complexity in this new context It is shown that there can be no simple relationship between distance to singularity and condition number, because the condition number can be computed in pol
TL;DR: In this paper, the existence of T-periodic positive solutions of the equation where f(t, x) lies between two lines of positive slope for large and positive x was studied.
Abstract: We study the existence of T-periodic positive solutions of the equationwhere f(t, .) has a singularity of repulsive type near the origin. Under the assumption that f(t, x) lies between two lines of positive slope for large and positive x, we find a non-resonance condition which predicts the existence of one T-periodic solution.Our main result gives a Fredholm alternative-like result for the existence of T-periodic positive solutions for
TL;DR: It is now apparent that the two principal models of turbulence are the extremes of a continuous family of (stable, attractive, hence «universal») multifractals characterized by Levy indices α=0 and 2, respectively.
Abstract: It is now apparent that the two principal models of turbulence (the «beta» and «lognormal» models) are the extremes of a continuous family of (stable, attractive, hence «universal») multifractals characterized by Levy indices α=0 and 2, respectively. Using a technique called double trace moment analysis, and turbulent velocity data, we empirically obtain α≃1.3±0.1: As has long been suspected, turbulence really is «in between» the β and lognormal models. This describes the entire hierarchy of singularities of the Navier-Stokes equations
TL;DR: This paper shows how to jump over a singularity by computing the next existing orthogonal polynomial by the block bordering method, and the resulting algorithm, called MRZ, is equivalent to the nongeneric BIODIR algorithm, but the derivation is much simpler.
Abstract: Lanczos type algorithms for solving systems of linear equations have their foundations in the theory of formal orthogonal polynomials and the method of moments which leads to a determinantal formula for their iterates. The various Lanczos type algorithms mainly differ by the way of computing the coefficients entering into the recurrence formulae. If the denominator in the formula for one of these coefficients is zero, then a breakdown occurs in the algorithm, and it must be stopped. Such a breakdown is in fact due to the non-existence of some orthogonal polynomial. In this paper we show how to jump over such a singularity by computing the next existing orthogonal polynomial by the block bordering method. The resulting algorithm, called MRZ, is equivalent to the nongeneric BIODIR algorithm (which is a look-ahead Lanczos type algorithm), but our derivation is much simpler.
TL;DR: A scheme which excises a region inside an apparent horizon containing the singularity through the use of a horizon-locking coordinate and a finite differencing which respects the causal structure of the spacetime.
Abstract: Progress in numerical relativity has been hindered for 30 years because of the difficulties of avoiding spacetime singularities in numerical evolution We propose a scheme which excises a region inside an apparent horizon containing the singularity Two major ingredients of the scheme are the use of a horizon-locking coordinate and a finite differencing which respects the causal structure of the spacetime Encouraging results of the scheme in the spherical collapse case are given
TL;DR: In this article, the structure and formation of naked singularities in selfsimilar gravitational collapse for an adiabatic perfect fluid were studied and conditions were obtained for the singularity to be either locally or globally naked and for the families of non-spacelike geodesics to terminate in past.
Abstract: We study the structure and formation of naked singularities in selfsimilar gravitational collapse for an adiabatic perfect fluid. Conditions are obtained for the singularity to be either locally or globally naked and for the families of non-spacelike geodesics to terminate at the singularity in past. This is shown to be a strong curvature naked singularity in a powerful sense and an interesting relationship is pointed out between positivity of energy and occurrence of naked singularity.
TL;DR: It is proved that the singularity-free solutions have a well-defined cylindrical symmetry and that they are generalizations of other singularity -free solutions obtained recently.
Abstract: We present a general class of solutions to Einstein's field equations with two spacelike commuting Killing vectors by assuming the separation of variables of the metric components. The solutions can be interpreted as inhomogeneous cosmological models. We show that the singularity structure of the solutions varies depending on the different particular choices of the parameters and metric functions. There exist solutions with a universal big-bang singularity, solutions with timelike singularities in the Weyl tensor only, solutions with singularities in both the Ricci and the Weyl tensors, and also singularity-free solutions. We prove that the singularity-free solutions have a well-defined cylindrical symmetry and that they are generalizations of other singularity-free solutions obtained recently.
TL;DR: Analytical expressions describing the singularity loci of simple planar three-degree-of-freedom parallel manipulators are derived using the analytical form of the Jacobian matrix, which is known to be singular when the manipulator is in a degenerate configuration.
TL;DR: The detailed numerical computations show various basic aspects of plate modeling in a concrete setting, including the boundary layer and corner singularities of the solution.
TL;DR: The Fermi-edge singularity in optical spectra is studied theoretically using the Tomonaga-Luttinger model for one-dimensional (1D) systems and the exponent is found to be independent of the hole dynamics in 1D, which is in striking contrast to the 2D and 3D cases.
Abstract: The Fermi-edge singularity in optical spectra is studied theoretically using the Tomonaga-Luttinger model for one-dimensional (1D) systems. Its critical exponent is obtained analytically for an arbitrary mass of a valence hole taking into account the electronic correlation. The exponent is found to be independent of the hole dynamics in 1D, which is in striking contrast to the 2D and 3D cases. Weak repulsive interaction among the conduction electrons sharpens the power-law peak in the edge spectrum.
TL;DR: In this paper, the singularity structure of the quark propagator obtained as a solution of the Schwinger-Dyson equation is studied, and it is found to be an entire function (except for an essential singularity at p2 = −∞; timelike in our metric): a property that may be indicative of confinement.
TL;DR: In this paper, simple solutions of Cauchy and hypersingular integrands in the gradient (flux or traction) boundary integral equation (BIE) were applied to linear elastic fracture analysis.
Abstract: The 'simple solutions' or 'indirect' method of analysing Cauchy and hypersingular integrands in the gradient (flux or traction) boundary integral equation (BIE) is applied to linear elastic fracture analysis. Because of the geometric singularity of the crack surface, application of the simple solutions formulas on the crack face requires integration over a temporary 'closure surface' rather than the remainder of the body. Closure surface constructions are exhibited for crack surfaces, allowing the gradient BIE to be applied as a constraint equation on a crack surface where the primary BIE is degenerate. Computational results are given for two benchmark fracture problems.