Vicinity Of Singularity Research Articles

Black hole singularity resolution in Wheeler–DeWitt quantum gravity

Ever since Hawking highlighted the puzzle of the existence of a singularity in the classical spacetime of general relativity, implying breakdown of all laws of physics living in the classical spacetime, singularity resolution became a problem of significantly high importance. This undesirable feature, indicating loss of predictability, has intrigued physicists over several decades. It has been hoped that the classical singularity would be resolved in a quantum theory of gravity. However, no appropriate wave function in the vicinity of the black hole singularity has been obtained so far to reach a definite conclusion.In this paper, we focus upon the interior of the Schwarzschild black hole, plagued by a non-removable classical singularity. We consider the interior spacetime of the black hole to be represented by the Kantowski–Sachs metric. Since in a quantum mechanical scenario, existence of spontaneous fluctuations of matter fields should not be ignored, we include a Klein–Gordon field in the system. Quantizing this simple gravity-matter model in the canonical scheme, we obtain the Wheeler–DeWitt equation and find an exact solution in the minisuperspace variables.We find that there exist three classes of solutions belonging to three different subregions of the eigenvalue space. Two of these classes of solutions admit the DeWitt criterion, a necessary condition for singularity resolution, implied by vanishing of the wave function at the singularity. These solutions are well-behaved and finite in the vicinity of the singularity and they indicate the existence of regular black holes in quantum gravity. In these classes of solutions, we further find that the expectation value of the Kretschmann operator is well-behaved and regular near the singularity, confirming a definite resolution to the puzzle of classical black hole singularity. On the other hand, there exists a small subregion in the eigenvalue space where the solution does not satisfy both conditions, the DeWitt criterion and finiteness of the Kretschmann expectation value at the singularity. This last class of solutions does not represent regular quantum black holes.

Uniformization and Constructive Analytic Continuation of Taylor Series

We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point, and additional analyticity properties. This situation occurs frequently in applications. The question of the optimal procedure was open, and we formulate it as a well-posed mathematical problem. Its solution leads to a practical method which provides dramatic accuracy improvements over existing techniques. Our procedure is based on uniformization of Riemann surfaces. As an application, we show that our procedure can be implemented for solutions of a wide class of nonlinear ODEs. We find a new uniformization method, which we use to construct the uniformizing maps needed for special functions, including solution of the Painlev'e equations P_I-P_V. We also introduce a new rigorous and constructive method of regularization, elimination of singularities whose position and type are known. If these are unknown, the same procedure enables a highly sensitive resonance method to determine the position and type of a singularity. In applications where less explicit information is available about the Riemann surface, our approach and techniques lead to new approximate, but still much more precise reconstruction methods than existing ones, especially in the vicinity of singularities, which are the points of greatest interest.

Numerical Valuation of European and American Options under Fractional Black-Scholes Model

In this paper, we investigate the numerical valuation of European and American options under the time fractional Black-Scholes model. We first apply a coordinate stretching transformation to the asset price so that the spatial region can focus on the vicinity of singularities, which are usually found in the payoff function. The radial basis function finite difference method is used for the spatial discretization, and the improved L1 method is used to deal with the reduced order of convergence for the nonsmooth initial data. We use the operator splitting method for solving the linear complementary problem of American options. The proposed scheme leads to a sparse linear system which is trivial to solve. Moreover, the stability of the proposed numerical scheme is analyzed using Fourier analysis. Numerical experiments demonstrate the accuracy and efficiency of the proposed method.

Global Steering for Control Moment Gyroscope Clusters Using Heuristic Variable Search Techniques

This paper addresses the problem of singularity avoidance in a four-control-moment-gyroscope cluster as used for the attitude control of a satellite. A global search algorithm is developed that adjusts the null motion added upon the singularity robust inverse steering law. Its principal characteristic is that it uses global information gathered from the whole maneuver, compared to most conventional techniques that consider only some local information, near the current gimbal configuration for the optimization. The method is implemented using a ternary tree structure, and a heuristic algorithm optimizes a cost function that depends on the manipulability index, the time spent in the vicinity of singularity, the quaternion error, and the gimbal rates. Specific measures are taken to decrease the execution time of the algorithm, even for long maneuvers, without the need of a visit histogram. In addition, the tuning of only two variables can drastically change the execution time and the computational resources needed. The numerical simulation used to evaluate the algorithm indicates that the elliptic singularity can sufficiently be avoided and the algorithm drives the system fast away from the singularity with an overall performance improvement.

Cost Minimizing Local Anisotropic Quad Mesh Refinement

AbstractQuad meshes as a surface representation have many conceptual advantages over triangle meshes. Their edges can naturally be aligned to principal curvatures of the underlying surface and they have the flexibility to create strongly anisotropic cells without causing excessively small inner angles. While in recent years a lot of progress has been made towards generating high quality uniform quad meshes for arbitrary shapes, their adaptive and anisotropic refinement remains difficult since a single edge split might propagate across the entire surface in order to maintain consistency. In this paper we present a novel refinement technique which finds the optimal trade‐off between number of resulting elements and inserted singularities according to a user prescribed weighting. Our algorithm takes as input a quad mesh with those edges tagged that are prescribed to be refined. It then formulates a binary optimization problem that minimizes the number of additional edges which need to be split in order to maintain consistency. Valence 3 and 5 singularities have to be introduced in the transition region between refined and unrefined regions of the mesh. The optimization hence computes the optimal trade‐off and places singularities strategically in order to minimize the number of consistency splits — or avoids singularities where this causes only a small number of additional splits. When applying the refinement scheme iteratively, we extend our binary optimization formulation such that previous splits can be undone if this prevents degenerate cells with small inner angles that otherwise might occur in anisotropic regions or in the vicinity of singularities. We demonstrate on a number of challenging examples that the algorithm performs well in practice.

Quantum singularities in self-similar spacetimes

Singularities, which are commonplace in general relativity, are indicated by causal geodesic incompleteness in otherwise maximal spacetimes. Can such singularities be healed (or “resolved”) quantum mechanically? Geodesics are in effect the paths of classical particles, which led Horowitz and Marolf to propose that they be replaced by quantum wave packets. Then a singularity is healed if the quantum wave operator can be shown to be essentially self-adjoint. We explore here a generalized Horowitz–Marolf approach for conformally static spacetimes in the context of the Klein–Gordon operator in the vicinity of singularities in the self-similar spacetimes introduced by Brady. We show that it fails the self-adjointness test for an entire generic class of spacetimes that have asymptotically power-law metric coefficients near the classically-singular origin, so the singularities in these spacetimes cannot be healed using quantum wave packets.

High order approximation to non-smooth multivariate functions

Common approximation tools return low-order approximations in the vicinities of singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form f=g+r+ where g,r∈CM+1(Rn) and the function r+ is defined byr+(y)={r(y),r(y)≥00,r(y)<0,∀y∈Rn. Given scattered (or uniform) data points X⊂Rn, we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities.

S -wave scattering length of a Gaussian potential

We provide accurate expressions for the $s$-wave scattering length for a Gaussian potential well in one, two and three spatial dimensions. The Gaussian potential is widely used as a pseudopotential in the theoretical description of ultracold atomic gases, where the $s$-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the $s$-wave scattering length from the parameters of the Gaussian but find that its accuracy is limited in the vicinity of singularities that result from the formation of new bound states. We then derive simple analytical expressions that capture the correct asymptotic behavior of the $s$-wave scattering length near the bound states. Expressions that are increasingly accurate in wide parameter regimes are found by a hierarchy of approximations that capture an increasing number of bound states. The small number of numerical coefficients that enter these expressions is determined from accurate numerical calculations. The approximate formulas combine the advantages of the numerical and approximate expressions, yielding an accurate and simple description from the weakly to the strongly interacting limit.

Volume-based phase stability testing at pressure and temperature specifications

Conventional phase equilibrium calculations at temperature and pressure specifications require the resolution of the equation of state (EoS). In volume-based calculations, the EoS must not be solved for volume, which is a primary variable. The tangent plane distance (TPD) function can be expressed in terms of mole numbers and volume or of component molar densities. The stationary points of the TPD function, as well as the location of the stability test limit locus (STLL) are different for the two formulations. A modified TPD function in terms of mole numbers and volume is proposed here. Using the block structure of the Hessian matrix, it is shown that Newton iterations in volume-based stability are inherently slower than those in the conventional PT stability, since the Hessian matrix is evaluated at a lower implicitness level. The minimization of several TPD functions (in volume-based and conventional stability testing) is analyzed using various sets of independent variables and scaling procedures. A modified Cholesky factorization and a two-stage line search procedure ensure a sequence of decreasing TPD functions in all cases. The proposed methods are tested for several mixtures, with emphasis on the vicinities of singularities (STLL and spinodal). With proper scaling, the modified Newton iterations are robust and converge very fast for most conditions and reasonably fast for very difficult conditions. The proposed algorithm is not model-dependent; any pressure explicit EoS can be used, provided the required partial derivatives are available.

Spectral anomalies and Gouy rotation around the singularity of ultrashort vortex pulses.

Spectral anomalies of femtosecond pulses with orbital angular momentum were studied in the vicinity of singularities. Bessel-Gauss (BG) beams were generated with mode-locked Ti:sapphire oscillators and dispersion-compensated diffractive axicons acting as spiral phase plates (SPPs). High-resolution two-dimensional spectral mapping was performed with a scanning fiber probe. Progressive rotation of the most pronounced features, known as "spectral eyes", in the maps of spectral moments was found at increasing propagation distance. The phenomenon is explained by a wavelength-dependent Gouy phase shift of interfering spectral components in the twisted wavefront. Spatial "spectral switching" was detected for few-cycle pulses. Possible improvements of selectivity are proposed.

Higher-order continuation method for the rigid-body kinematic design of compliant mechanisms

Compliant mechanisms are of great interest in precision engineering. In this paper, we propose a higher-order continuation method to help their rigid-body kinematic design. The method helps to investigate the choice of a mechanism configuration through the whole exploration of the workspace, and eases the kinematic analysis to avoid, or take advantage of, the vicinity of kinematic singularities. Such approach is relevant for planar and quasi-planar mechanisms that can be obtained with micro-manufacturing processes adapted to precision applications. The higher-order continuation method allows for a direct and accurate plotting of the input–output relationship of any mechanism by considering only its geometrical closed-loop equations, i.e. without the complex derivation of any analytical model. We show that these plots, called bifurcation diagrams, reveal essential information such as the joint velocity profile and the presence of singular configurations. Moreover, the continuous and accurate computation of the mechanism configuration in the vicinity of singularities provides detailed information about the kinematic behavior of the mechanism in its extreme positions. For the design of compliant mechanisms, the designer can advantageously use the bifurcation diagrams to evaluate the relevance of the selected mechanism, then to identify a configuration in order to obtain desired kinematic properties without the derivation of the inverse kinematic model (IKM) or the direct kinematic model (DKM). The method is exemplified with a 3 universal-joint and 3 spherical-joint mechanism (3-US), the IKM and DKM of which cannot be derived analytically. The latter has a large workspace and special kinematic behaviors consisting of a screw-like motion and a platform gyration, which have not been studied before and could lead to novel compliant devices.

Hierarchical Constraint‐Based Singularity Avoidance for Multi‐Axis Robots

AbstractA generalized reflexive approach for Hierarchical Constraint‐Based Singularity Avoidance (HCB‐SA) is proposed and demonstrated for the multi‐axis robot control of a six Degrees Of Freedom (DOF) hybrid manipulator system. The concept utilizes a dynamic adaptation of virtual constraints by introducing virtual damped actuation on the velocity control level and an anisotropic reflexive trajectory deflection depending on the robot constraints on the position control level. Redundant or low priority DOFs can be used to minimize the pose error in the more important DOFs reflexively without calculating new trajectories. Accordingly, the end‐effector can be safely controlled in the vicinity of singularities and all constraints in the task and joint space can be surely hold. Furthermore, the proposed cascaded feedback control with the generalized HCB‐SA algorithm is able to react on the presence of external disturbances which is validated using software‐in‐the‐loop simulation on the real‐time control system. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Simple timing generation along workspace paths for nonredundant robotic limbs

This article focuses on the relation between workspace path geometry and the timing along the path, obtained via simple timing generators yielding constant end-link speed motion, natural motion, and two types of globally optimized joint velocity motions. The generators are designed within the Singularity-Consistent framework developed originally to tackle motion control in the vicinity of kinematic singularities. A comparative study highlights how performance expressed in terms of various kinematic and dynamic criteria of local (peak joint velocity and torque) and global (joint velocity/torque uniformity and total mechanical power) nature is influenced by the curvature of the path image in configuration space and by the vicinity of singular configurations. Results from simulations with a simple planar 2R limb and a spatial 3R positioning limb following linear and circular paths are presented.

Robust and efficient Trust-Region based stability analysis and multiphase flash calculations

Multiphase flash calculations and phase stability analysis are central in compositional reservoir and chemical process simulators. For instance, in some simulations, a huge amount of phase equilibrium calculations is required (the most important part of the computational effort). Moreover, a single failure may cause significant error propagations leading to false solutions. Thus, it is imperative that calculation algorithms are efficient and highly robust. The most difficult regions in mixture phase envelopes are in the vicinity of singularities: critical points for flash calculations and the stability test limit locus for stability analysis. For these conditions, all algorithms have difficulties to converge. Traditionally, a number of successive substitution iterations (SSI) are performed before switching to the second-order Newton method (many SSI iteration may be required before switch very close to singularities). The Trust-Region method has the advantage of performing a Newton step whenever the Hessian is definite positive; otherwise, the Trust-Region corrects the Hessian matrix by adding a diagonal element to make it positive definite, thus a descent direction is guaranteed. The Trust-Region limits the solution within a trust-radius, which is updated automatically at each iteration level, depending on the quality of the quadratic approximation. If the function is convex, the trust-radius enables larger changes in iteration variables, otherwise restricted steps are used to ensure a progress towards the solution. The proposed Trust-Region algorithm, as well as a hybrid methodology that combines SSI, Newton and Trust-Region steps, are tested for multiphase flash calculations and stability analysis on a variety of mixtures involving hydrocarbon components, carbon dioxide and hydrogen sulfide, exhibiting complicated phase envelopes. The proposed method compares favorably to the widely used SSI-Newton methods with various independent variables. The more difficult a test point is, the more spectacular the algorithm acts from both efficiency and reliability perspectives.

Formation of curvature singularities on the interface between dielectric liquids in a strong vertical electric field

The nonlinear dynamics of the interface between two deep dielectric fluids in the presence of a vertical electric field is studied. We consider the limit of a strong external electric field where electrostatic forces dominate over gravitational and capillary forces. The nonlinear integrodifferential equations for the interface motion are derived under the assumption of small interfacial slopes. It is shown in the framework of these equations that, in the generic case, the instability development leads to the formation of root singularities at the interface in a finite time. The interfacial curvature becomes infinite at singular points, while the slope angles remain relatively small. The curvature is negative in the vicinity of singularities if the ratio of the permittivities of the fluids exceeds the inverse ratio of their densities, and it is positive in the opposite case (we consider that the lower fluid is heavier than the upper one). In the intermediate case, the interface evolution equations describe the formation and sharpening of dimples at the interface. The results obtained are applicable for the description of the instability of the interface between two magnetic fluids in a vertical magnetic field.

Delaminated multilayered plates under uniaxial extension. Part II: Efficient layerwise mesh strategy for the prediction of delamination onset

In the first part (Saeedi et al., 2011) of this two-part paper, a layerwise stress model, called the LS1 model, has been extended to the analysis of delaminated multilayered plates subjected to uniaxial extension. The LS1 analytical solutions of general delaminated multilayered plates have been derived and compared to three-dimensional finite element method (3D-FEM) solutions. It has been proved that there is a good agreement between the LS1 and 3D-FEM models except very near singularities (free edges, crack tips,…). In order to overcome the drawback of the LS1 model in the vicinity of singularities, a refinement approach, called the refined LS1 is proposed in this part. Based on an irregular layerwise mesh, the refined LS1 model is applied to the prediction of the delamination onset in angle-ply composite laminates. The comparison between the refined LS1 and 3D-FE models reveals an excellent agreement in terms of interlaminar stresses and strain energy release rate, even very close to singularities. The proposed refined LS1 model can be used as an accurate and very efficient model for the prediction of initiation and propagation of delamination in multilayered plates under uniaxial extension using stress based or energy release rate based criteria.

Delaminated multilayered plates under uniaxial extension. Part I: Analytical analysis using a layerwise stress approach

The aim of this two-part paper is to propose an efficient and accurate alternative to the computationally expensive three-dimensional finite element method (3D-FEM), for analyzing delaminated multilayered plates under uniaxial extension. Many of the existing models deal with the analysis of multilayered structures only in the non-delaminated state. The first part of the present study extends the application of a layerwise stress model, called the LS1 model, to delaminated multilayered plates subjected to uniaxial extension. The analytical LS1 solutions are derived for general non-delaminated and delaminated multilayers and compared to 3D finite element solutions. The comparison gives a good agreement between the LS1 and 3D-FE models except near singularities (free edges, crack tips,…). In order to overcome this drawback, a refinement approach, called the refined LS1, is presented in Part II and applied to angle-ply rectangular composite laminates. The comparison between the refined LS1 and 3D-FE models reveals an excellent agreement, even in the vicinity of singularities, in terms of interlaminar stresses and strain energy release rate. The main conclusion of the second part is that the proposed refined LS1 model can be used as an accurate and very efficient model for evaluating the interfacial stress fields as well as the strain energy release rate in multi-delamination problems.

Null cosmological singularities and free strings: II

In arXiv:0909.4731 [hep-th], we argued that the free string lightcone Schrodinger wavefunctional in the vicinity of null Kasner-like cosmological singularities has nonsingular time-dependence if the Kasner exponents satisfy certain relations. These backgrounds are anisotropic plane waves with singularities. We first show here that only certain singularities admit a Rosen-Kasner frame with exponents satisfying relations leading to a wavefunctional with nonsingular time-dependence. Then we build on the (Rosen) description further and study various physical observables for a time-dependent harmonic oscillator toy model and then the free string, reconciling this with the corresponding description in the conventional plane wave variables. We find that observables containing no time derivatives are identical in these variables while those with time derivatives are different. Various free string observables are still divergent, perhaps consistent with string oscillator states becoming light in the vicinity of the singularity.

Null cosmological singularities and free strings

We continue exploring free strings in the background of null Kasner-like cosmological singularities, following K. Narayan, arXiv:0904.4532. We study the free string Schrodinger wave functional along the lines of K. Narayan, arXiv:0807.1517. We find the wave functional to be nonsingular in the vicinity of singularities whose Kasner exponents satisfy certain relations. We compare this with the description in other variables. We then study certain regulated versions of these singularities where the singular region is replaced by a substringy but nonsingular region and study the string spectra in these backgrounds. The string modes can again be solved for exactly, giving some insight into how string oscillator states get excited near the singularity.

Pressing a rigid plastic surface by a wedge-shaped stamp under the Coulomb-Mohr yield condition

The problem of treating a surface with a wedge-shaped stamp is considered using the model of an ideal rigid plastic body. The strain fields in the vicinity of singularities of the displacement velocity field (on the discontinuity lines of the displacement velocities and at the center of the fan of characteristics) are investigated taking into account irreversible compressibility.