Showing papers on "Singularity published in 1998"
TL;DR: In this article, a physically motivated regularization of the Euler equations is proposed to allow topological transitions to occur smoothly, where the sharp interface is replaced by a narrow transition layer across which the fluids may mix.
Abstract: One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.
878 citations
TL;DR: In this paper, the authors introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria for porous composite materials, initially by studying the stress states of the so-called rank 2 layered materials, then, on the basis of the theoretical study of the rank 2 microstructures, they propose an empirical model that extends the power penalized stiffness model (also called SIMP for Solid Isotropic Microstructure with Penalization for intermediate densities).
Abstract: We introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria. We first consider relevant stress criteria for porous composite materials, initially by studying the stress states of the so-called rank 2 layered materials. Then, on the basis of the theoretical study of the rank 2 microstructures, we propose an empirical model that extends the power penalized stiffness model (also called SIMP for Solid Isotropic Microstructure with Penalization for inter-mediate densities). In a second part, solution aspects of topology problems are considered. To deal with the so-called ‘singularity’ phenomenon of stress constraints in topology design, an ϵ-constraint relaxation of the stress constraints is used. We describe the mathematical programming approach that is used to solve the numerical optimization problems, and show results for a number of example applications. © 1998 John Wiley & Sons, Ltd.
638 citations
01 Jan 1998
TL;DR: A discrete technique of the Schwarz alternating method is presented, to combine the Ritz-Galerkin and finite element methods, well suited for solving singularity problems in parallel.
Abstract: A discrete technique of the Schwarz alternating method is presented in this last chapter, to combine the Ritz-Galerkin and finite element methods. This technique is well suited for solving singularity problems in parallel, and requires a little more computation for large overlap of subdomains. The convergence rate of the iterative procedure, which depends upon overlap of subdomains, will be studied. Also a balance strategy will be proposed to couple the iteration number with the element size used in the FEM. For the crack-infinity problem of singularity the total CPU time by the technique in this chapter is much less than that by the nonconforming combination in Chapter 12.
389 citations
TL;DR: A detailed study of the singularity theorems is presented in this article, where the authors discuss the plausibility and reasonability of their hypotheses, applicability and implications of singularity theories, as well as the theorem itself.
Abstract: A detailed study of the singularity theorems is presented. I discuss the plausibility and reasonability of their hypotheses, the applicability and implications of the theorems, as well as the theorems themselves. The consequences usually extracted from them, some of them without the necessary rigour, are widely and carefully analysed with many clarifying examples and alternative views.
283 citations
TL;DR: In this paper, the authors used techniques from the theory of ODEs and also from inverse scattering theory to obtain a variety of results on the regularity and support properties of the equilibrium measure for logarithmic potentials on the finite interval?1, 1], in the presence of an external fieldV.
268 citations
TL;DR: In this article, the authors consider field theories arising from a large number of D3-branes near singularities in F-theory and compute, using their conjectured string theory duals, their large N spectrum of chiral primary operators.
Abstract: We consider field theories arising from a large number of D3-branes near singularities in F-theory. We study the theories at various conformal points, and compute, using their conjectured string theory duals, their large N spectrum of chiral primary operators. This includes, as expected, operators of fractional conformal dimensions for the theory at Argyres-Douglas points. Additional operators, which are charged under the (sometimes exceptional) global symmetries of these theories, come from the 7-branes. In the case of a D4 singularity we compare our results with field theory and find agreement for large N. Finally, we consider deformations away from the conformal points, which involve finding new supergravity solutions for the geometry produced by the 3-branes in the 7-brane background. We also discuss 3-branes in a general background.
267 citations
TL;DR: A number of second order maps are presented, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable, and a more sensitive integrability test is proposed using the growth of the degree of its iterates.
Abstract: We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the analysis of the complexity (``algebraic entropy'') of the map using the growth of the degree of its iterates: integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems.
261 citations
TL;DR: In this article, the authors consider field theories arising from a large number of D3-branes near singularities in F-theory and compute, using their conjectured string theory duals, their large $N$ spectrum of chiral primary operators.
Abstract: We consider field theories arising from a large number of D3-branes near singularities in F-theory. We study the theories at various conformal points, and compute, using their conjectured string theory duals, their large $N$ spectrum of chiral primary operators. This includes, as expected, operators of fractional conformal dimensions for the theory at Argyres-Douglas points. Additional operators, which are charged under the (sometimes exceptional) global symmetries of these theories, come from the 7-branes. In the case of a $D_4$ singularity we compare our results with field theory and find agreement for large $N$. Finally, we consider deformations away from the conformal points, which involve finding new supergravity solutions for the geometry produced by the 3-branes in the 7-brane background. We also discuss 3-branes in a general background.
234 citations
TL;DR: In this article, it is shown that the saddle cannot close in finite time and it cannot be faster than a double exponential in time, and the same results hold for incompressible 2D and 3D Euler vorticity equations.
Abstract: The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is "no" in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations. In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak [7] suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: "If the level set topology in the temperature field for the 2D quasi-geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle, then no finite time singularity is possible." Numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle. There is a naturally associated notion of simple hyperbolic saddle breakdown. The main theorem we present in this paper shows that such breakdown cannot occur in finite time. We also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time. Using the same techniques, we see that analogous results hold for incompressible 2D and 3D Euler. These results were announced in [9], but with a slight difference in the definition of a simple hyperbolic saddle. The definition given in Section 4 generalizes the one given in the announcement. See also Constantin [4], discussed in Section 7, Remark 5 below. The whole work described in this paper is basically part of the author's thesis. I am particularly grateful to my thesis advisor Charles Fefferman for his attention, support, guidance and advice. I am indebted to D. Christodoulou and P. Constantin for helpful corrections and suggestions. I wish to thank A. Majda for suggesting the subject and E. Tabak for discussions and com-
193 citations
TL;DR: In this article, an initial value problem for the conformal Einstein equations is formulated such that the data and equations are regular, space-like and null infinity have a finite representation, with their structure and location known a priori, and the setting relies entirely on general properties of conformal structures.
188 citations
TL;DR: In this paper, the geometry and elasticity of the singularity on a crumpled elastic sheet were analyzed in terms of a free-boundary contact problem, and an analytical solution was given for the universal shape of a developable cone that characterizes singularity far from the tip.
Abstract: We analyze the geometry and elasticity of the crescentlike singularity on a crumpled elastic sheet. We give a physical realization of this in terms of a free-boundary contact problem. An analytical solution is given for the universal shape of a developable cone that characterizes the singularity far from the tip, and some of its predictions are qualitatively verified experimentally. We also give a scaling relation for the core size, defined as the region close to the tip of the cone where the sheet is not developable.
23 Jun 1998
TL;DR: This work describes a novel 2-D Scaled Prismatic Model (SPM) for figure registration that has fewer singularity problems and does not require detailed knowledge of the 3-D kinematics.
Abstract: We analyze the use of kinematic constraints for articulated object tracking. Conditions for the occurrence of singularities in 3-D models are presented and their effects on tracking are characterized We describe a novel 2-D Scaled Prismatic Model (SPM) for figure registration. In contrast to 3-D kinematic models, the SPM has fewer singularity problems and does not require detailed knowledge of the 3-D kinematics. We fully characterize the singularities in the SPM and illustrate tracking through singularities using synthetic and real examples with 3-D and 2-D models. Our results demonstrate the significant benefits of the SPM in tracking with a single source of video.
TL;DR: In this paper, the Fuchsian algorithm was used to construct singular solutions of Einstein's equations which belong to the class of Gowdy spacetimes and provided precise asymptotics at the singularity which is Kasner-like.
Abstract: We use the Fuchsian algorithm to construct singular solutions of Einstein's equations which belong to the class of Gowdy spacetimes. The solutions have the maximum number of arbitrary functions. Special cases correspond to polarized or other known solutions. The method provides precise asymptotics at the singularity, which is Kasner-like. All of these solutions are asymptotically velocity-dominated. The results account for the fact that solutions with velocity parameter uniformly greater than one are not observed numerically. They also provide a justification of formal expansions proposed by Grubisic and Moncrief.
TL;DR: In this article, Brane Box Models of intersecting NS and D5 branes are mapped to D3 branes at 3/Γ singularities and viceversa, in a setup which gives rise to N = 1 supersymmetric gauge theories in four dimensions.
Abstract: Brane Box Models of intersecting NS and D5 branes are mapped to D3 branes at 3/Γ singularities and viceversa, in a setup which gives rise to N = 1 supersymmetric gauge theories in four dimensions. The Brane Box Models are constructed on a two-torus. The map is interpreted as T-duality along the two directions of the torus. Some Brane Box Models contain NS fivebranes winding around (p,q) cycles in the torus, and our method provides the geometric T-dual to such objects. An amusing aspect of the mapping is that T-dual configurations are calculated using D = 4 N = 1 field theory data. The mapping to the singularity picture allows the geometrical interpretation of all the marginal couplings in finite field theories. This identification is further confirmed using the AdS/CFT correspondence for orbifold theories. The AdS massless fields coupling to the marginal operators in the boundary appear as stringy twisted sectors of S5/Γ. The mapping for theories which are non-finite requires the introduction of fractional D3 branes in the singularity picture.
TL;DR: In this article, a unified analytical description of classical bulk solids and fluids is obtained, predicting correctly the major features of their equations of state and freezing parameters as obtained by simulations, on the basis of the fundamental-measure free energy functional for hard spheres and thermodynamic perturbation theory.
Abstract: On the basis of the fundamental-measure free energy functional for hard spheres and thermodynamic perturbation theory, a unified analytical description of classical bulk solids and fluids is obtained, predicting correctly the major features of their equations of state and freezing parameters as obtained by simulations. The fundamentally different fluid and solid asymptotic high density expansions for the potential energy, featuring a static-lattice Madelung term and the harmonic 3/2k B T correction, on one hand, and a fluid Madelung energy with a ˜T 3/5 thermal energy correction, on the other, both originate from the same singularity in the hard sphere free energy functional.
Posted Content•
TL;DR: In this article, it was shown that the saddle cannot close in finite time and it cannot be faster than a double exponential in time, and the same results hold for incompressible 2D and 3D Euler vorticity equations.
Abstract: The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is ``no'' in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations.
In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: ``If the level set topology in the temperature field for the 2D quasi-geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle, then no finite time singularity is possible.''
Numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle. There is a naturally associated notion of simple hyperbolic saddle breakdown. The main theorem we present in this paper shows that such breakdown cannot occur in finite time. We also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time. Using the same techniques, we see that analogous results hold for incompressible 2D and 3D Euler.
TL;DR: In this article, the authors derived the asymptotic laws and their leading-asymptotic correction formulas for the motion of a tagged particle near a glass-transition singularity.
Abstract: Within mode-coupling theory for structural relaxation in simple systems, the asymptotic laws and their leading-asymptotic correction formulas are derived for the motion of a tagged particle near a glass-transition singularity. These analytic results are compared with numerical ones of the equations of motion evaluated for a tagged hard sphere moving in a hard-sphere system. It is found that the long-time part of the two-step relaxation process for the mean-squared displacement can be characterized by the a-relaxation scaling law and von Schweidler’s power-law decay, while the critical-decay regime is dominated by the corrections to the leading power-law behavior. For parameters of interest for the interpretations of experimental data, the corrections to the leading asymptotic laws for the non-Gaussian parameter are found to be so large that the leading asymptotic results are altered qualitatively by the corrections. Results for the non-Gaussian parameter are shown to follow qualitatively the findings reported in the molecular-dynamics-simulations work by Kob and Andersen @Phys. Rev. E 51, 4626 ~1995!#. @S1063-651X~98!09609-3#
Posted Content•
TL;DR: In this article, the category of coherent sheaves on the minimal resolution of the Kleinian singularity associated to a finite subgroup G of SL(2) was derived and applied to the Euler-characteristic version of the Hall algebra.
Abstract: We describe the derived category of coherent sheaves on the minimal resolution of the Kleinian singularity associated to a finite subgroup G of SL(2). Then, we give an application to the Euler-characteristic version of the Hall algebra of the category of coherent sheaves on an algebraic surface.
TL;DR: In this paper, a method for finding and classifying all the singularities of an arbitrary non-redundant mechanism is presented. But this method is based on the velocity-equation formulation of kinematic singularity and the singularity classification first introduced in (Zlatanov et al., 1994-1,2).
TL;DR: In this article, Brane Box Models of intersecting NS and D5 branes are mapped to D3 branes at C^3/Gamma orbifold singularities and vise versa, in a setup which gives rise to N=1 supersymmetric gauge theories in four dimensions.
Abstract: Brane Box Models of intersecting NS and D5 branes are mapped to D3 branes at C^3/Gamma orbifold singularities and vise versa, in a setup which gives rise to N=1 supersymmetric gauge theories in four dimensions. The Brane Box Models are constructed on a two-torus. The map is interpreted as T-duality along the two directions of the torus. Some Brane Box Models contain NS fivebranes winding around (p,q) cycles in the torus, and our method provides the geometric T-dual to such objects. An amusing aspect of the mapping is that T-dual configurations are calculated using D=4 N=1 field theory data. The mapping to the singularity picture allows the geometrical interpretation of all the marginal couplings in finite field theories. This identification is further confirmed using the AdS/CFT correspondence for orbifold theories. The AdS massless fields coupling to the marginal operators in the boundary appear as stringy twisted sectors of S^5/Gamma. The mapping for theories which are non-finite requires the introduction of fractional D3 branes in the singularity picture.
Book•
13 May 1998
TL;DR: A survey of singularity theory by means of differential forms can be found in this article, where the authors introduce the Gauss-Manin connection on the vanishing cohomology of a singularity, and draw on the work of Brieskorn and Steenbrink to calculate this connection.
Abstract: This 1998 book is both an introduction to, and a survey of, some topics of singularity theory; in particular the studying of singularities by means of differential forms. Here some ideas and notions that arose in global algebraic geometry, namely mixed Hodge structures and the theory of period maps, are developed in the local situation to study the case of isolated singularities of holomorphic functions. The author introduces the Gauss–Manin connection on the vanishing cohomology of a singularity, that is on the cohomology fibration associated to the Milnor fibration, and draws on the work of Brieskorn and Steenbrink to calculate this connection, and the limit mixed Hodge structure. This will be an excellent resource for all researchers whose interests lie in singularity theory, and algebraic or differential geometry.
TL;DR: This article is a guide to the literature on existence theorems for the Einstein equations which also draws attention to open problems in the field and global results for solutions with symmetry are discussed.
Abstract: This article is a guide to the literature on existence theorems for the Einstein equations which also draws attention to open problems in the field. The local in time Cauchy problem, which is relatively well understood, is treated first. Next global results for solutions with symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. This is followed by a survey of global results in the case of small data and results on constructing spacetimes with given singularity structure. The article ends with some miscellaneous topics connected with the main theme.
TL;DR: In this paper, the divergence of the stray field in corners and its consequence for micromagnetic calculations was studied numerically in two dimensions for high-anisotropy materials, and it was shown that no atomistic theory has to be invoked because the singularity is smoothed out already within micromagnetics.
TL;DR: In this paper, the singularity in generic gravitational collapse is explored analytically and numerically in spatially inhomogeneous cosmological space-times, with a convenient choice of variables.
Abstract: A longstanding conjecture by Belinskii, Khalatnikov and Lifshitz that the singularity in generic gravitational collapse is spacelike, local and oscillatory is explored analytically and numerically in spatially inhomogeneous cosmological space–times. With a convenient choice of variables, it can be seen analytically how nonlinear terms in Einstein's equations control the approach to the singularity and cause oscillatory behavior. The analytic picture requires the drastic assumption that each spatial point evolves toward the singularity as an independent spatially homogeneous universe. In every case, detailed numerical simulations of the full Einstein evolution equations support this assumption.
TL;DR: In this paper, it was shown that string theory on a conifold singularity can be described by supersymmetric gauge theory, which can be expressed in terms of supersymmetry.
Abstract: Just as parallel threebranes on a smooth manifold are related to string theory on $AdS_5\times {\bf S}^5$, parallel threebranes near a conical singularity are related to string theory on $AdS_5\times X_5$, for a suitable $X_5$. For the example of the conifold singularity, for which $X_5=(SU(2)\times SU(2))/U(1)$, we argue that string theory on $AdS_5\times X_5$ can be described by a certain ${\cal N}=1$ supersymmetric gauge theory which we describe in detail.
TL;DR: In this paper, two online singularity avoidance schemes are proposed to find an alternate path of the manipulator near singularity to keep the actuator forces always within their capacities, which is shown to yield a minimum deviation of the PPM from the specified path but it requires a high computation time.
TL;DR: In this article, the Laplace transform is applied to remove the time-dependent variable in the diffusion equation for nonharmonic initial conditions, which gives rise to a non-homogeneous modified Helmholtz equation which is solved by the method of fundamental solutions.
Abstract: The Laplace transform is applied to remove the time-dependent variable in the diffusion equation For non-harmonic initial conditions this gives rise to a non-homogeneous modified Helmholtz equation which we solve by the method of fundamental solutions To do this a particular solution must be obtained which we find through a method suggested by Atkinson To avoid costly Gaussian quadratures, we approximate the particular solution using quasi-Monte-Carlo integration which has the advantage of ignoring the singularity in the integrand The approximate transformed solution is then inverted numerically using Stehfest's algorithm Two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2-D and 3-D © 1998 John Wiley & Sons, Ltd
01 Jan 1998
TL;DR: In this paper, a method for the formulation of boundary integral equations with kernels of any order of singularity is outlined, and a semianalytical technique of general applicability for direct evaluation of singular integrals, as required in any boundary element implementation, is described in detail.
Abstract: Boundary integral equations with strongly singular and hypersingular kernels can be very useful in many elds of applied mechanics. In this paper two basic aspects are addressed. First, a method for the formulation of boundary integral equations with kernels of any order of singularity is outlined. The method never employs arbitrary interpretations in the Cauchy principal value or nite part sense. Essentially it amounts to show that no unbounded terms ultimately arise if the limiting process is properly performed. Second, a semianalytical technique of general applicability for the direct evaluation of singular integrals, as required in any boundary element implementation, is described in detail. The theory is supported by some numerical tests. Further developments and applications are also mentioned.
TL;DR: In this article, it was shown that a globally naked, shell-focusing singularity can occur at the center from relativistically high-density, isentropic, and time symmetric initial data if ≥ 0.01$ within the numerical accuracy.
Abstract: The final fate of the spherically symmetric collapse of a perfect fluid which follows the \ensuremath{\gamma}-law equation of state and adiabatic condition is investigated. Full general relativistic hydrodynamics is solved numerically using a retarded time coordinate, the so-called observer time coordinate. Thanks to this coordinate, the causal structure of the resultant space-time is automatically constructed. Then, it is found that a globally naked, shell-focusing singularity can occur at the center from relativistically high-density, isentropic, and time symmetric initial data if $\ensuremath{\gamma}\ensuremath{\lesssim}1.01$ within the numerical accuracy. The result is free from the assumption of self-similarity. The upper limit of \ensuremath{\gamma} with which a naked singularity can occur from generic initial data is consistent with the result of Ori and Piran based on the assumption of self-similarity.
TL;DR: In this article, the cosmological time function of a spacetime manifold has been shown to be globally hyperbolic and can be connected to the initial singularity by a geodesic ray that maximizes the distance to the singularity.
Abstract: Let be a time-oriented Lorentzian manifold and d the Lorentzian distance on M. The function is the cosmological time function of M, where as usual p< q means that p is in the causal past of q. This function is called regular iff for all q and also along every past inextendible causal curve. If the cosmological time function of a spacetime is regular it has several pleasant consequences: (i) it forces to be globally hyperbolic; (ii) every point of can be connected to the initial singularity by a rest curve (i.e. a timelike geodesic ray that maximizes the distance to the singularity); (iii) the function is a time function in the usual sense; in particular, (iv) is continuous, in fact, locally Lipschitz and the second derivatives of exist almost everywhere.