Showing papers on "Singularity published in 2014"

New strategies for some issues of numerical manifold method in simulation of crack propagation

Abstract: SUMMARY Aiming to solve, in a unified way, continuous and discontinuous problems in geotechnical engineering, the numerical manifold method introduces two covers, namely, the mathematical cover and the physical cover. In order to reach the goal, some issues in the simulation of crack propagation have to be solved, among which are the four issues to be treated in this study: (1) to reduce the rank deficiency induced by high degree polynomials as local approximation, a new variational principle is formulated, which suppresses the gradient-dependent DOFs; (2) to evaluate the integrals with singularity of 1/r, a new numerical quadrature scheme is developed, which is simpler but more efficient than the existing Duffy transformation; (3) to analyze kinked cracks, a sign convention for argument in the polar system at the crack tip is specified, which leads to more accurate results in a simpler way than the existing mapping technique; and (4) to demonstrate the mesh independency of numerical manifold method in handling strong singularity, a mesh deployment scheme is advised, which can reproduce all singular locations of the crack with regard to the mesh. Corresponding to the four issues, typical examples are given to demonstrate the effectiveness of the proposed schemes. Copyright © 2013 John Wiley & Sons, Ltd.

6d Conformal Matter

Abstract: A single M5-brane probing G, an ADE-type singularity, leads to a system which has G x G global symmetry and can be viewed as "bifundamental" (G,G) matter. For the A_N series, this leads to the usual notion of bifundamental matter. For the other cases it corresponds to a strongly interacting (1,0) superconformal system in six dimensions. Similarly, an ADE singularity intersecting the Horava-Witten wall leads to a superconformal matter system with E_8 x G global symmetry. Using the F-theory realization of these theories, we elucidate the Coulomb/tensor branch of (G,G') conformal matter. This leads to the notion of fractionalization of an M5-brane on an ADE singularity as well as fractionalization of the intersection point of the ADE singularity with the Horava-Witten wall. Partial Higgsing of these theories leads to new 6d SCFTs in the infrared, which we also characterize. This generalizes the class of (1,0) theories which can be perturbatively realized by suspended branes in IIA string theory. By reducing on a circle, we arrive at novel duals for 5d affine quiver theories. Introducing many M5-branes leads to large N gravity duals.

Orbifolds of M-strings

Abstract: We consider M-theory in the presence of M parallel M5-branes prob- ing a transverse AN−1 singularity. This leads to a superconformal theory with (1,0) supersymmetry in six dimensions. We compute the supersymmetric partition func- tion of this theory on a two-torus, with arbitrary supersymmetry preserving twists, using the topological vertex formalism. Alternatively, we show that this can also be obtained by computing the elliptic genus of an orbifold of recently studied M-strings. The resulting 2d theory is a (4,0) supersymmetric quiver gauge theory whose Higgs branch corresponds to strings propagating on the moduli space of SU(N) M−1 instan- tons on R 4 where the right-moving fermions are coupled to a particular bundle.

Integrable discrete PT symmetric model.

Abstract: An exactly solvable discrete $PT$ invariant nonlinear Schr\"odinger-like model is introduced. It is an integrable Hamiltonian system that exhibits a nontrivial nonlinear $PT$ symmetry. A discrete one-soliton solution is constructed using a left-right Riemann-Hilbert formulation. It is shown that this pure soliton exhibits unique features such as power oscillations and singularity formation. The proposed model can be viewed as a discretization of a recently obtained integrable nonlocal nonlinear Schr\"odinger equation.

Potentially singular solutions of the 3D axisymmetric Euler equations

Abstract: The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(ts − t)−2.46, where ts ∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ2 = 0.003505 at which time the vorticity amplifies by more than (3 × 108)-fold and the maximum mesh resolution exceeds (3 × 1012)2. The vorticity vector is observed to maintain four significant digits throughout the computations.

Explicit de Sitter flux vacua for global string models with chiral matter

Abstract: We address the open question of performing an explicit stabilisation of all closed string moduli (including dilaton, complex structure and Kahler moduli) in fluxed type IIB Calabi-Yau compactifications with chiral matter. Using toric geometry we construct Calabi-Yau manifolds with del Pezzo singularities. D-branes located at such singularities can support the Standard Model gauge group and matter content or some close extensions. In order to control complex structure moduli stabilisation we consider Calabi-Yau manifolds which exhibit a discrete symmetry that reduces the effective number of complex structure moduli. We calculate the corresponding periods in the symplectic basis of invariant three-cycles and find explicit flux vacua for concrete examples. We compute the values of the flux superpotential and the string coupling at these vacua. Starting from these explicit complex structure solutions, we obtain AdS and dS minima where the Kahler moduli are stabilised by a mixture of D-terms, non-perturbative and perturbative α ′ corrections as in the LARGE Volume Scenario. In the considered example the visible sector lives at a dP6 singularity which can be higgsed to the phenomenologically interesting class of models at the dP3 singularity.

Critical exponents of the 3d Ising and related models from conformal bootstrap

Abstract: The constraints of conformal bootstrap are applied to investigate a set of conformal field theories in various dimensions. The prescriptions can be applied to both unitary and non unitary theories allowing for the study of the spectrum of low-lying primary operators of the theory. We evaluate the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity for 2 ≤ D ≤ 6. Likewise we obtain the scaling dimensions of six scalars and four spinning operators for the 3d critical Ising model. Our findings are in agreement with existing results to a per mill precision and estimate several new exponents.

The derivation of swarming models: Mean-field limit and Wasserstein distances

Abstract: These notes are devoted to a summary on the mean-field limit of large ensembles of interacting particles with applications in swarming models. We first make a summary of the kinetic models derived as continuum versions of second order models for swarming. We focus on the question of passing from the discrete to the continuum model in the Dobrushin framework. We show how to use related techniques from fluid mechanics equations applied to first order models for swarming, also called the aggregation equation. We give qualitative bounds on the approximation of initial data by particles to obtain the mean-field limit for radial singular (at the origin) potentials up to the Newtonian singularity. We also show the propagation of chaos for more restricted set of singular potentials.

Universality of Tip Singularity Formation in Freezing Water Drops

Abstract: A drop of water deposited on a cold plate freezes into an ice drop with a pointy tip. While this phenomenon clearly finds its origin in the expansion of water upon freezing, a quantitative description of the tip singularity has remained elusive. Here we demonstrate how the geometry of the freezing front, determined by heat transfer considerations, is crucial for the tip formation. We perform systematic measurements of the angles of the conical tip, and reveal the dynamics of the solidification front in a Hele-Shaw geometry. It is found that the cone angle is independent of substrate temperature and wetting angle, suggesting a universal, self-similar mechanism that does not depend on the rate of solidification. We propose a model for the freezing front and derive resulting tip angles analytically, in good agreement with the experiments.

Monotone Hurwitz Numbers and the HCIZ Integral

Abstract: In this article, we study the topological expansion of the Harish-Chandra-Itzykson-Zuber matrix model. We prove three types of results concerning the free energy of the HCIZ model. First, at the exact level, we express each derivative of the HCIZ free energy as an absolutely convergent series in inverse powers of the ensemble dimension. The coefficients in this series are generating polynomials for a desymmetrization of the double Hurwitz numbers which we call monotone double Hurwitz numbers. Second, we prove that the genus-specific generating functions for the monotone double Hurwitz numbers are convergent power series with a common dominant singularity at the critical point 2/27. The analytic functions defined by these series are candidate orders for a conjectural asymptotic expansion of the free energy postulated by Matytsin. Finally, we prove that under a non-vanishing hypothesis on the partition function the HCIZ free energy converges to the generating function for genus zero monotone double Hurwitz numbers uniformly on compact subsets of a complex domain.

Singularity structure of maximally supersymmetric scattering amplitudes

Abstract: We present evidence that loop amplitudes in maximally supersymmetric (N=4) Yang-Mills theory (SYM) beyond the planar limit share some of the remarkable structures of the planar theory. In particular, we show that through two loops, the four-particle amplitude in full N=4 SYM has only logarithmic singularities and is free of any poles at infinity—properties closely related to uniform transcendentality and the UV finiteness of the theory. We also briefly comment on implications for maximal (N=8) supergravity theory (SUGRA).

Distinguishing black holes from naked singularities through their accretion disc properties

Abstract: We show that, in principle, a slowly evolving gravitationally collapsing perfect fluid cloud can asymptotically settle to a static spherically symmetric equilibrium configuration with a naked singularity at the center. We consider one such asymptotic final configuration with a finite outer radius, and construct a toy model in which it is matched to a Schwarzschild exterior geometry. We examine the properties of circular orbits in this model. We then investigate the observational signatures of a thermal accretion disc in this spacetime, comparing them with the signatures expected for a disc around a black hole of the same mass. Several notable differences emerge. A disc around the naked singularity is much more luminous than one around an equivalent black hole. Also, the disc around the naked singularity has a spectrum with a high frequency power law segment that carries a major fraction of the total luminosity. Thus, at least some naked singularities can, in principle, be distinguished observationally from the black holes of the same mass. We discuss the possible implications of these results.

Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation

Abstract: Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10^(12))^2 near the point of the singularity, we are able to advance the solution up to tau_2 = 0.003505 and predict a singularity time of t(s) approximate to 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector. and observing a (3 x 10^8)-fold increase in the maximum vorticity parallel to omega parallel to(infinity). The numerical data are checked against all major blowup/non-blowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.

On the Finite-Time Splash and Splat Singularities for the 3-D Free-Surface Euler Equations

Abstract: We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time “splash” (or “splat”) singularity first introduced in Castro et al. (Splash singularity for water waves, http://arxiv.org/abs/1106.2120v2, 2011), wherein the evolving 2-D hypersurface, the moving boundary of the fluid domain, self-intersects at a point (or on surface). Such singularities can occur when the crest of a breaking wave falls unto its trough, or in the study of drop impact upon liquid surfaces. Our approach is founded upon the Lagrangian description of the free-boundary problem, combined with a novel approximation scheme of a finite collection of local coordinate charts; as such we are able to analyze a rather general set of geometries for the evolving 2-D free-surface of the fluid. We do not assume the fluid is irrotational, and as such, our method can be used for a number of other fluid interface problems, including compressible flows, plasmas, as well as the inclusion of surface tension effects.

Conical Kähler–Einstein Metrics Revisited

Abstract: In this paper we introduce the “interpolation–degeneration” strategy to study Kahler–Einstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By “interpolation” we show the angles in (0, 2π] that admit a conical Kahler–Einstein metric form a connected interval, and by “degeneration” we determine the boundary of the interval in some important cases. As a first application, we show that there exists a Kahler–Einstein metric on \({\mathbb{P}^2}\) with cone singularity along a smooth conic (degree 2) curve if and only if the angle is in (π/2, 2π]. When the angle is 2π/3 this proves the existence of a Sasaki–Einstein metric on the link of a three dimensional A2 singularity, and thus answers a question posed by Gauntlett–Martelli–Sparks–Yau. As a second application we prove a version of Donaldson’s conjecture about conical Kahler–Einstein metrics in the toric case using Song–Wang’s recent existence result of toric invariant conical Kahler–Einstein metrics.

Topological strings and quantum spectral problems

Abstract: We consider certain quantum spectral problems appearing in the study of local Calabi-Yau geometries. The quantum spectrum can be computed by the Bohr-Sommerfeld quantization condition for a period integral. For the case of small Planck constant, the periods are computed perturbatively by deformation of the Ω background parameters in the Nekrasov-Shatashvili limit. We compare the calculations with the results from the standard perturbation theory for the quantum Hamiltonian. There have been proposals in the literature for the non-perturbative contributions based on singularity cancellation with the perturbative contributions. We compute the quantum spectrum numerically with some high precisions for many cases of Planck constant. We find that there are also some higher order non-singular non-perturbative contributions, which are not captured by the singularity cancellation mechanism. We fix the first few orders formulas of such corrections for some well known local Calabi-Yau models.

A general mechanism to generate three limit cycles in planar Filippov systems with two zones

Abstract: Discontinuous piecewise linear systems with two zones are considered. A general canonical form that includes all the possible configurations in planar linear systems is introduced and exploited. It is shown that the existence of a focus in one zone is sufficient to get three nested limit cycles, independently on the dynamics of the another linear zone. Perturbing a situation with only one hyperbolic limit cycle, two additional limit cycles are obtained by using an adequate parametric sector of the unfolding of a codimension-two focus-fold singularity.

Singularity-free dislocation dynamics with strain gradient elasticity

Abstract: The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.

Quantum correlations and entanglement in far-from-equilibrium spin systems

Abstract: By applying complementary analytic and numerical methods, we investigate the dynamics of spin-$\frac{1}{2} XXZ$ models with variable-range interactions in arbitrary dimensions. The dynamics we consider is initiated from uncorrelated states that are easily prepared in experiments; it can be equivalently viewed as either Ramsey spectroscopy or a quantum quench. Our primary focus is the dynamical emergence of correlations and entanglement in these far-from-equilibrium interacting quantum systems: We characterize these correlations by the entanglement entropy, concurrence, and squeezing, which are inequivalent measures of entanglement corresponding to different quantum resources. In one spatial dimension, we show that the time evolution of correlation functions manifests a nonperturbative dynamic singularity. This singularity is characterized by a universal power-law exponent that is insensitive to small perturbations. Explicit realizations of these models in current experiments using polar molecules, trapped ions, Rydberg atoms, magnetic atoms, and alkaline-earth and alkali-metal atoms in optical lattices, along with the relative merits and limitations of these different systems, are discussed.

Analytical computation of gravity effects for polyhedral bodies

Abstract: On the basis of recent analytical results we derive new formulas for computing the gravity effects of polyhedral bodies which are expressed solely as function of the coordinates of the vertices of the relevant faces. We thus prove that such formulas exhibit no singularity whenever the position of the observation point is not aligned with an edge of a face. In the opposite case, the contribution of the edge to the potential to its first-order derivative and to the diagonal entries of the second-order derivative is deemed to be zero on the basis of some claims which still require a rigorous mathematical proof. In contrast with a common statement in the literature, it is proved that only the off-diagonal entries of the second-order derivative of the potential do exhibit a noneliminable singularity when the observation point is aligned with an edge of a face. The analytical provisions on the range of validity of the derived formulas have been fully confirmed by the Matlab $$^{\textregistered }$$ program which has been coded and thoroughly tested by computing the gravity effects induced by real asteroids at arbitrarily placed observation points.

Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators

Abstract: A classical pseudodifferential operator P on ℝn satisfies the μ-transmission condition relative to a smooth open subset Ω when the symbol terms have a certain twisted parity on the normal to ∂Ω. As shown recently by the author, this condition assures solvability of Dirichlet-type boundary problems for P in full scales of Sobolev spaces with a singularity dμ−k, d(x)= dist(x,∂Ω). Examples include fractional Laplacians (−Δ)a and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type, or, more generally, nonlocal type. It is also shown how problems with data on ℝn∖Ω reduce to problems supported on Ω¯, and how the so-called “large” solutions arise. Moreover, the results are extended to general function spaces Fp,qs and Bp,qs, including Holder–Zygmund spaces B∞,∞s. This leads to optimal Holder estimates, e.g., for Dirichlet solutions of (−Δ)au=f∈L∞(Ω), u∈daCa(Ω¯) when 0

A Grassmann path from AdS(3) to flat space

Abstract: We show that interpreting the inverse AdS(3) radius 1/l as a Grassmann variable results in a formal map from gravity in AdS(3) to gravity in flat space. The underlying reason for this is the fact that ISO(2, 1) is the Inonu-Wigner contraction of SO(2, 2). We show how this works for the Chern-Simons actions, demonstrate how the general (Banados) solution in AdS(3) maps to the general flat space solution, and how the Killing vectors, charges and the Virasoro algebra in the Brown-Henneaux case map to the corresponding quantities in the BMS3 case. Our results straightforwardly generalize to the higher spin case: the recently constructed flat space higher spin theories emerge automatically in this approach from their AdS counterparts. We conclude with a discussion of singularity resolution in the BMS gauge as an application.

Finiteness of algebraic fundamental groups

Abstract: We show that the algebraic local fundamental group of any Kawamata log terminal singularity as well as the algebraic fundamental group of the smooth locus of any log Fano variety are finite.

The Zel'dovich approximation: key to understanding cosmic web complexity

Abstract: We describe how the dynamics of cosmic structure formation defines the intricate geometric structure of the spine of the cosmic web. The Zel'dovich approximation is used to model the backbone of the cosmic web in terms of its singularity structure. The description by Arnold et al. in terms of catastrophe theory forms the basis of our analysis. This two-dimensional analysis involves a profound assessment of the Lagrangian and Eulerian projections of the gravitationally evolving four-dimensional phase-space manifold. It involves the identification of the complete family of singularity classes, and the corresponding caustics that we see emerging as structure in Eulerian space evolves. In particular, as it is instrumental in outlining the spatial network of the cosmic web, we investigate the nature of spatial connections between these singularities. The major finding of our study is that all singularities are located on a set of lines in Lagrangian space. All dynamical processes related to the caustics are concentrated near these lines. We demonstrate and discuss extensively how all 2D singularities are to be found on these lines. When mapping this spatial pattern of lines to Eulerian space, we find a growing connectedness between initially disjoint lines, resulting in a percolating network. In other words, the lines form the blueprint for the global geometric evolution of the cosmic web.

Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem.

Abstract: In this paper, for both the sharp front surface quasi-geostrophic equation and the Muskat problem, we rule out the "splash singularity" blow-up scenario; in other words, we prove that the contours evolving from either of these systems cannot intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem. Our result confirms the numerical simulations in earlier work, in which it was shown that the curvature blows up because the contours collapse at a point. Here, we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of earlier work in which squirt singularities are ruled out; in this case, a positive volume of fluid between the contours cannot be ejected in finite time.

Spectral instability of characteristic boundary layer flows

Abstract: In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number: $R \to \infty$ Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity The formal construction of approximate modes is well-documented in physics literature, going back to the work of Heisenberg, CC Lin, Tollmien, Drazin and Reid, but a rigorous construction requires delicate mathematical details, involving for instance a treatment of primitive Airy functions and singular solutions Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of $e^{t/\sqrt {R}}$ A new, operator-based approach is introduced, avoiding to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators

Analytic test configurations and geodesic rays

Abstract: Starting with the data of a curve of singularity types, we use the Legendre transform to construct weak geodesic rays in the space of locally bounded metrics on an ample line bundle L over a compact manifold. Using this we associate weak geodesics to suitable filtrations of the algebra of sections of L. In particular this works for the natural filtration coming from an algebraic test configuration, and we show how this recovers the weak geodesic ray of Phong-Sturm.

Quantum Raychaudhuri equation

Abstract: We compute quantum corrections to the Raychaudhuri equation by replacing classical geodesics with quantal (Bohmian) trajectories and show that they prevent focusing of geodesics and the formation of conjugate points We discuss implications for the Hawking-Penrose singularity theorems and curvature singularities

On the Effective Metric of a Planck Star

Abstract: Spacetime metrics describing `non-singular' black holes are commonly studied in the literature as effective modification to the Schwarzschild solution that mimic quantum gravity effects removing the central singularity. Here we point out that to be physically plausible, such metrics should also incorporate the 1-loop quantum corrections to the Newton potential and a non-trivial time delay between an observer at infinity and an observer in the regular center. We present a modification of the well-known Hayward metric that features these two properties. We discuss bounds on the maximal time delay imposed by conditions on the curvature, and the consequences for the weak energy condition, in general violated by the large transversal pressures introduced by the time delay.