Showing papers on "Singularity published in 1996"

Singularity patterns in a chemotaxis model

Abstract: The authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time. Such a finite-time blow-up of solutions implies chemotactic collapse, namely, concentration of species to form sporae. The model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity, which has the form ut=Δu−χ(uv), 0=Δv+(u−1), on ΩR2, where Ω is an open disc with no-flux (homogeneous Neumann) boundary conditions. The initial conditions are continuous functions u(x,0)=u0(x)≥0, v(x,0)=v0(x)≥0 for xΩ. Under these conditions, the authors prove there exists a radially symmetric solution u(r,t) which blows up at r=0, t=T<∞. A specific description of such a solution is presented. The authors also discuss the strong similarity between the chemotactic model they study and the classical Stefan problem.

Summing up Dirichlet Instantons.

Abstract: We investigate quantum corrections to the moduli space for hypermultiplets for the type IIA string near a conifold singularity. We find a unique quantum deformation based on symmetry arguments which is consistent with a recent conjecture. The correction can be interpreted as an infinite sum coming from multiple wrappings of the Euclidean Dirichlet branes around the vanishing cycle.

Matter From Geometry

Abstract: We provide a local geometric description of how charged matter arises in type IIA, M-theory, or F-theory compactifications on Calabi-Yau manifolds. The basic idea is to deform a higher singularity into a lower one through Cartan deformations which vary over space. The results agree with expectations based on string dualities.

Notes on the eigensystem of magnetohydrodynamics

Abstract: The eigenstructure of the equations governing one-dimensional ideal magnetohy-drodynamics is examined, motivated by the wish to exploit it for construction of high-resolution computational algorithms. The results are given in simple forms that avoid indeterminacy or degeneracy whenever possible. The unavoidable indeterminacy near the magnetosonic (or triple umbilic) state is analysed and shown to cause no difficulty in evaluating a numerical flux function. The structure of wave paths close to this singularity is obtained, and simple expressions are presented for the structure coefficients that govern wave steepening.

On the reconstruction of the vibro‐acoustic field over the surface enclosing an interior space using the boundary element method

Abstract: The vibrational velocity, sound pressure, and acoustic power on the vibrating boundary comprising an enclosed space are reconstructed by the boundary element method based on the measured field pressures. The singular value decomposition is used to obtain the inverse solution in the least‐square sense and to express the acoustic modal expansion between the measurement and source fields. In general, such an inverse operation has been considered an ill‐posed problem having a divergence phenomenon involved with extremely small measurement errors. The ill‐conditioned nature of the acoustic inverse problem is caused by the singularity of the transfer matrix which produces nonradiating wave components. In order to minimize the singularity and to also reduce the number of measurement points, optimal measurement positions are determined by the effective independence method. Regularization methods are used to stabilize the reconstructed field by suppressing nonradiating components resulting in the singular transfer...

Lectures on Finite Precision Computations

Abstract: Foreword Iain S. Duff Preface General Presentation Notations. Part I. Computability in Finite Precision: Well-Posed Problems Approximations Convergence in Exact Arithmetic Computability in Finite Precision Gaussian Elimination Forward Error Analysis The Influence of Singularities Numerical Stability in Exact Arithmetic Computability in Finite Precision for Iterative and Approximate Methods The Limit of Numerical Stability in Finite Precision Arithmetically Robust Convergence The Computed Logistic Bibliographical Comments. Part II. Measures of Stability for Regular Problems: Choice of Data and Class of Perturbations Choice of Norms: Scaling Conditioning of Regular Problems Simple Roots of Polynomials Factorizations of a Complex Matrix Solving Linear Systems Functions of a Square Matrix Concluding Remarks Bibliographical Comments. Part III. Computation in the Neighbourhood of a Singularity: Singular Problems Which are Well-Posed Condition Numbers of Holder-Singularities Computability of Ill-Posed Problems Singularities of z ----> A - zI Distances to Singularity Unfolding of Singularity Spectral Portraits Bibliographical Comments. Part IV. Arithmetic Quality of Reliable Algorithms: Forward and Backward Analyses Backward Error Quality of Reliable Software Formulae for Backward Errors Influence of the Class of Perturbations Iterative Refinement for Backward Stability Robust Reliability and Arithmetic Quality Bibliographical Comments. Part V. Numerical Stability in Finite Precision: Iterative and Approximate Methods Numerical Convergence of Iterative Solvers Stopping Criteria in Finite Precision Robust Convergence The Computed Logistic Revisited Care of Use Bibliographical Comments. PartVI. Software Tools for Round-Off Error Analysis in Algorithms. A Historical Perspective The Assessment of the Quality of the Numerical Software Backward Error Analysis in Libraries Sensitivity Analysis Interval Analysis Probabilisitc Models Computer Algebra Bibliographical Comments. Part VII. The Toolbox PRECISE for Computer Experimentation. What is PRECISE? Module for Backward Error Analysis Sample Size Backward Analysis with PRECISE Dangerous Border and Unfolding of a Singularity Summary of Module 1 Bibliographical Comments. Part VIII. Experiments with PRECISE. Format of the Examples Backward Error Analysis for Linear Systems Computer Unfolding of Singularity Dangerous Border and Distance to Singularity Roots of Polynomials Eigenvalue Problems Conclusion Bibliographical Comments. Part IX. Robustness to Nonnormality. Nonnormality and Spectral Instability Nonnormality in Physics and Technology Convergence of Numerical Methods in Exact Arithmetic Influence on Numerical Software Bibliographical Comments. Part V. Qualitative Computing. Sensitivity and Pseudosolutions for F (x) = y: Pseudospectra of Matrices Pseudozeroes of Polynomials Divergence Portrait for the Complex Logistic Iteration Qualitative Computation of a Jordan Form Beyond Linear Perturbation Theory Bibliographical Comments. Part XI. More Numerical Illustrations with PRECISE: Annex: The Toolbox PRECISE for MATLAB Index Bibliography.

The final fate of spherical inhomogeneous dust collapse

Abstract: We examine, in a general manner, the role played by the initial density and velocity distributions in the gravitational collapse of a spherically symmetric inhomogeneous dust cloud. Such a collapse is described by the Tolman - Bondi metric which has two free functions: the `mass function' and the `energy function', which are determined by the initial density and velocity profiles of the cloud. The collapse can end in either a black hole or a naked singularity, depending on the initial parameters characterizing these profiles. In the marginally bound case, we find that the collapse ends in a naked singularity if the leading non-vanishing derivative of the density at the centre is either the first one or the second one. If the first two derivatives are zero, and the third derivative non-zero, the singularity could either be naked or covered, depending on a quantity determined by the third derivative and the central density. If the first three derivatives are zero, the collapse ends in a black hole. In particular, the classic result of Oppenheimer and Snyder, that homogeneous dust collapse leads to a black hole, is recovered as a special case. Analogous results are found when the cloud is not marginally bound, and also for the case of a cloud starting from rest. A condition on the initial density profile is given for the singularity to be globally naked. We also show how the strength of the naked singularity depends on the density and velocity distribution. Our analysis generalizes and simplifies the earlier works of Christodoulou and Newman by dropping the assumption of evenness of density functions. It turns out that relaxing this assumption allows for a smooth transition from the naked singularity phase to the black hole phase, and also allows for the occurrence of strong curvature naked singularities.

A local-to-global singularity theorem for quantum field theory on curved space-time

Abstract: We prove that if a reference two-point distribution of positive type on a time orientable curved space-time (CST) satisfies a certain condition on its wave front set (the “classP M,g condition”) and if any other two-point distribution (i) is of positive type, (ii) has the same antisymmetric part as the reference modulo smooth function and (iii) has the same local singularity structure, then it has the same global singularity structure. In the proof we use a smoothing, positivity-preserving pseudo-differential operator the support of whose symbol is restricted to a certain conic region which depends on the wave front set of the reference state. This local-to-global theorem, together with results published elsewhere, leads to a verification of a conjecture by Kay that for quasi-free states of the Klein-Gordon quantum field on a globally hyperbolic CST, the local Hadamard condition implies the global Hadamard condition. A counterexample to the local-to-global theorem on a strip in Minkowski space is given when the classP M,g condition is not assumed.

Singularities in inflationary cosmology: A Review

Abstract: We review here some recent results that show that inflationary cosmological models must contain initial singularities. We also present a new singularity theorem. The question of the initial singularity re-emerges in inflationary cosmology because inflation is known to be generically futureeternal, It is natural to ask, therefore, if inflationary models can be continued into the infinite past in a non-singular way. The results that we discuss show that the answer to the question is “no.” This means that we cannot use inflation as a way of avoiding the question of the birth of the Universe. We also argue that our new theorem suggests — in a sense that we explain in the paper — that the Universe cannot be infinitely old.

String and M-Theory Cosmological Solutions with Ramond Forms

Abstract: A general framework for studying a large class of cosmological solutions of the low-energy limit of type II string theory and of M-theory, with non-trivial Ramond form fields excited, is presented. The framework is applicable to spacetimes decomposable into a set of flat or, more generally, maximally symmetric spatial subspaces, with multiple non-trivial form fields spanning one or more of the subspaces. It is shown that the corresponding low-energy equations of motion are equivalent to those describing a particle moving in a moduli space consisting of the scale factors of the subspaces together with the dilaton. The choice of which form fields are excited controls the potential term in the particle equations. Two classes of exact solutions are given, those corresponding to exciting only a single form and those with multiple forms excited which correspond to Toda theories. Although typically these solutions begin or end in a curvature singularity, there is a subclass with positive spatial curvature which appears to be singularity free. Elements of this class are directly related to certain black p-brane solutions.

Detailed structure and propagation of three-dimensional detonations

Abstract: Self-sustaining detonations exhibit a three-dimensional structure within the reaction zone which is much more complex than one-dimensional classical theory. While the structure of two-dimensional detonations is now fairly well understood, the details of the three-dimensional structure remain largely unknown. Numerical simulation, performed with the reactive Euler equations coupled to a single-step chemistry model, yields insight into the three-dimensional reaction zone structure. Simulations in a square channel show that, in the absence of wall losses, the three-dimensional transverse wave spacing is the same as in two dimensions. However, the transverse wave structure is much more intricate. Two perpendicular modes exist in this channel, and they are approximately one-quarter of a period out of phase. This phase shift accounts for the slapping wave phenomenon observed in experiments. The slapping wave imprints on smoke foils occur when the transverse wave collides with the wall. The curvature of the imprints is explained by the curvature of the slapping wave, which is convex toward the wall. The interaction of the two transverse waves results in a vorticity field that is much more complex than in two dimensions. In two dimensions, vorticity originates from a point singularity, which is now replaced by a line singularity that forms closed loops interconnecting the two sets of transverse waves. The interconnection changes after collisions, creating cuts and free edges on the sheet structure which otherwise remains globally interconnected. It is the vorticity in the part of the sheet initiating or ending along a free edge that rolls up into rings. The rings entrain unburned fluid behind the Mach stem, as in two dimensions. The vorticity system provides a strong coupling mechanism between the two orthogonal sets of transverse waves.

Optimal spectral bandwidth for long memory

Abstract: For long range dependent time series with a spectral singularity at frequency zero, a theory for optimal bandwidth choice in non-parametric analysis ofthe singularity was developed by Robinson (1991b). The optimal bandwidths are described and compared with those in case of analysis of a smooth spectrum. They are also analysed in case of fractional ARIMA models and calculated as a function of the self similarity parameter in some special cases. Feasible data dependent approximations to the optimal bandwidth are discussed.

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Abstract: Explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features: degeneracy, quasilinearity, full nonlinearity, and singularities. In particular, the equation of “motion by mean curvature” is included. The schemes are monotone and consistent, so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems. In addition, an intriguing new type of nonlocal problem is analyzed which is related to the schemes, and another very different sort of approximation is presented as well.

Product integration for Volterra integral equations of the second kind with weakly singular kernels

Abstract: We introduce a new numerical approach for solving Volterra integral equations of the second kind when the kernel contains a mild singularity. We give a convergence result. We also present numerical examples which show the performance and efficiency of our method.

Nonlinear wave equations

Abstract: Linear wave propagation local and global existence singularity formation solitons and inverse scattering perturbation methods general relativity.

Symmetric singularity formation in lubrication-type equations for interface motion

Abstract: Fourth-order degenerate diffusion equations arise in a “lubrication approximation” of a thin film or neck driven by surface tension. Numerical studies of the lubrication equation (LE) $h_t + ( h^n h_{xxx} )_x = 0$ with various boundary conditions indicate that singularity formation in which $h( x( t ),t ) \to 0$ occurs for small enough n with “anomalous” or “second type” scaling inconsistent with usual dimensional analysis.This paper considers locally symmetric or even singularities in the (LE) and in the modified lubrication equation (MLE) $h_t + h^n h_{xxxx} = 0$. Both equations have the property that entropy bounds forbid finite-time singularities when n is sufficiently large. Power series expansions for local symmetric similarity solutions are proposed for equation (LE) with $n < 1$ and (MLE) for all $n \in \mathbb{R}$. In the latter case, special boundary conditions that force singularity formation are required to produce singularities when n is large. Matching conditions at higher-order terms in the...

Tree level string cosmology

Abstract: In this paper we examine the classical evolution of a cosmological model derived from the low-energy tree-level limit of a generic string theory The action contains the metric, dilaton, central charge and an antisymmetric tensor field We show that with a homogeneous and isotropic metric, allowing spatial curvature, there is a formal equivalence between this system and a scalar field minimally coupled to Einstein gravity in a spatially flat metric We refer to this system as the shifted frame and using it we describe the full range of cosmological evolution that this model can exhibit We show that generic solutions begin (or end) with a singularity As the system approaches a singularity the dilaton becomes large and loop corrections will become important {copyright} {ital 1996 The American Physical Society}

Computation of periodic Green’s functions of Stokes flow

Abstract: Methods of computing periodic Green’s functions of Stokes flow representing the flow due to triply-, doubly-, and singly-periodic arrays of three-dimensional or two-dimensional point forces are reviewed, developed, and discussed with emphasis on efficient numerical computation. The standard representation in terms of Fourier series requires a prohibitive computational effort for use with singularity and boundary-integral-equation methods; alternative representations based on variations of Ewald’s summation method involving various types of splitting between physical and Fourier space with partial sums that decay in a Gaussian or exponential manner, allow for efficient numerical computation. The physical changes undergone by the flow in deriving singly- and doubly- periodic Green’s functions from their triply-periodic counterparts are considered.

The onset of instability in unsteady boundary-layer separation

Abstract: The process of unsteady two-dimensional boundary-layer separation at high Reynolds number is considered. Solutions of the unsteady non-interactive boundary-layer equations are known to develop a generic separation singularity in regions where the pressure gradient is prescribed and adverse. As the boundary layer starts to separate from the surface, however, the external pressure distribution is altered through viscous-inviscid interaction just prior to the formation of the separation singularity; hitherto this has been referred to as the first interactive stage. A numerical solution of this stage is obtained here in Lagrangian coordinates. The solution is shown to exhibit a high-frequency inviscid instability resulting in an immediate finite-time breakdown of this stage. The presence of the instability is confirmed through a linear stability analysis. The implications for the theoretical description of unsteady boundary-layer separation are discussed, and it is suggested that the onset of interaction may occur much sooner than previously thought.

Singularity formation in Hele–Shaw bubbles

Abstract: We provide numerical and analytic evidence for the formation of a singularity driven only by surface tension in the mathematical model describing a two‐dimensional Hele–Shaw cell with no air injection. Constantin and Pugh have proved that no such singularity is possible if the initial shape is close to a circle; thus we show that their result is not true in general. Our evidence takes the form of direct numerical simulation of the full problem, including a careful assessment of the effects of limited spatial resolution, and comparison of the full problem with the lubrication approximation.

Geometric interpretation and classification of global solutions in generalized dilaton gravity.

Abstract: Two dimensional gravity with torsion is proved to be equivalent to special types of generalized 2d dilaton gravity. E.g. in one version, the dilaton field is shown to be expressible by the extra scalar curvature, constructed for an independent Lorentz connection corresponding to a nontrivial torsion. Elimination of that dilaton field yields an equivalent torsionless theory, nonpolynomial in curvature. These theories, although locally equivalent exhibit quite different global properties of the general solution. We discuss the example of a (torsionless) dilaton theory equivalent to the R 2 + T 2 –model. Each global solution of this model is shown to split into a set of global solutions of generalized dilaton gravity. In contrast to the theory with torsion the equivalent dilaton one exhibits solutions which are asymptotically flat in special ranges of the parameters. In the simplest case of ordinary dilaton gravity we clarify the well kown problem of removing the Schwarzschild singularity by a field redefinition.

Standardized complex and logarithmic eigensolutions for n -material wedges and junctions

Abstract: The Airy stress eigenfunction expansion of Williams [1] has been used to obtain simple expressions for the angular variations of the stress and displacement fields for n-material wedges and junctions subjected to inplane loading. This formulation applies to real and complex roots, as well as the special transition case giving rise to r −ω singular behavior. The asymptotic behavior of the general problem is similar to that of the bi-material interface crack. In the case of real roots, the stress and displacement expressions can be determined to within a multiplicative real constant (amplification), while for the complex case, the fields are determined to within a multiplicative complex constant (amplification plus rotation). Because of the rotation in the complex case, there are an infinite number of equivalent ways to express the angular variations (eigenfunctions) of the stress and displacement fields. Therefore, the fields are standardized in terms of ‘generalized stress intensity factors’ that are consistent with the bi-material interface crack and the homogeneous crack problems. As in the bi-material crack problem, for the complex case there are two stress intensity factors for each admissible order of the stress singularity. For specific n-material wedges and junctions, a small variation of material properties and/or geometry can change the eigenvalues from a pair of complex conjugate roots to two distinct real roots or vice-versa. An r −ω singularity associated with a nonseparable solution in υ and θ exists at this point of bifurcation. Such behavior requires an adjustment in the standard eigenfunction approach to insure bounded stress intensity factors. The proper form of the solution is given both at and near this special material combination, and the smooth transition of the eigenfunctions as the roots change from real to complex is demonstrated in the results. Additional eigenfunction results are provided for particular cases of 2 and 3-material wedges and junctions.