Showing papers on "Singularity published in 2017"

Singularity: Scientific containers for mobility of compute.

Abstract: Here we present Singularity, software developed to bring containers and reproducibility to scientific computing. Using Singularity containers, developers can work in reproducible environments of their choosing and design, and these complete environments can easily be copied and executed on other platforms. Singularity is an open source initiative that harnesses the expertise of system and software engineers and researchers alike, and integrates seamlessly into common workflows for both of these groups. As its primary use case, Singularity brings mobility of computing to both users and HPC centers, providing a secure means to capture and distribute software and compute environments. This ability to create and deploy reproducible environments across these centers, a previously unmet need, makes Singularity a game changing development for computational science.

Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation

Abstract: A reaction-diffusion problem with a Caputo time derivative of order $\alpha\in (0,1)$ is considered. The solution of such a problem is shown in general to have a weak singularity near the initial time $t=0$, and sharp pointwise bounds on certain derivatives of this solution are derived. A new analysis of a standard finite difference method for the problem is given, taking into account this initial singularity. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.

Looking for a bulk point

Abstract: We consider Lorentzian correlators of local operators. In perturbation theory, singularities occur when we can draw a position-space Landau diagram with null lines. In theories with gravity duals, we can also draw Landau diagrams in the bulk. We argue that certain singularities can arise only from bulk diagrams, not from boundary diagrams. As has been previously observed, these singularities are a clear diagnostic of bulk locality. We analyze some properties of these perturbative singularities and discuss their relation to the OPE and the dimensions of double-trace operators. In the exact nonperturbative theory, we expect no singularity at these locations. We prove this statement in 1+1 dimensions by CFT methods.

Analytic description of singularities in Gowdy spacetimes

Abstract: We use Fuchsian Reduction 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 Grubi\v{s}i\'c and Moncrief.

4d \( \mathcal{N}=1 \) from 6d (1, 0)

Abstract: We study the geometry of 4d $$ \mathcal{N}=1 $$ SCFT’s arising from compactification of 6d (1, 0) SCFT’s on a Riemann surface. We show that the conformal manifold of the resulting theory is characterized, in addition to moduli of complex structure of the Riemann surface, by the choice of a connection for a vector bundle on the surface arising from flavor symmetries in 6d. We exemplify this by considering the case of 4d $$ \mathcal{N}=1 $$ SCFT’s arising from M5 branes probing ℤ k singularity compactified on a Riemann surface. In particular, we study in detail the four dimensional theories arising in the case of two M5 branes on ℤ 2 singularity. We compute the conformal anomalies and indices of such theories in 4d and find that they are consistent with expectations based on anomaly and the moduli structure derived from the 6 dimensional perspective.

Super-Coulombic atom–atom interactions in hyperbolic media

Abstract: Dipole-dipole interactions, which govern phenomena such as cooperative Lamb shifts, superradiant decay rates, Van der Waals forces and resonance energy transfer rates, are conventionally limited to the Coulombic near-field. Here we reveal a class of real-photon and virtual-photon long-range quantum electrodynamic interactions that have a singularity in media with hyperbolic dispersion. The singularity in the dipole-dipole coupling, referred to as a super-Coulombic interaction, is a result of an effective interaction distance that goes to zero in the ideal limit irrespective of the physical distance. We investigate the entire landscape of atom-atom interactions in hyperbolic media confirming the giant long-range enhancement. We also propose multiple experimental platforms to verify our predicted effect with phonon-polaritonic hexagonal boron nitride, plasmonic super-lattices and hyperbolic meta-surfaces as well. Our work paves the way for the control of cold atoms above hyperbolic meta-surfaces and the study of many-body physics with hyperbolic media.

$E_8$ instantons on type-A ALE spaces and supersymmetric field theories

Abstract: We consider the 6d superconformal field theory realized on M5-branes probing the $E_8$ end-of-the-world brane on the deformed and resolved $\mathbb{C}^2/\mathbb{Z}_k$ singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the $T^2$ compactification, and the star-shaped quiver obtained as the mirror of the $T^3$ compactification.

Finite-Size Scaling of a First-Order Dynamical Phase Transition: Adaptive Population Dynamics and an Effective Model.

Abstract: We analyze large deviations of the time-averaged activity in the one-dimensional Fredrickson-Andersen model, both numerically and analytically. The model exhibits a dynamical phase transition, which appears as a singularity in the large deviation function. We analyze the finite-size scaling of this phase transition numerically, by generalizing an existing cloning algorithm to include a multicanonical feedback control: this significantly improves the computational efficiency. Motivated by these numerical results, we formulate an effective theory for the model in the vicinity of the phase transition, which accounts quantitatively for the observed behavior. We discuss potential applications of the numerical method and the effective theory in a range of more general contexts.

Investigation of Negative-Sequence Injection Capability of Cascaded H-Bridge Converters in Star and Delta Configuration

Abstract: The aim of this paper is to investigate the ability of cascaded H-bridge STATic COMpensator in star and delta configuration to exchange negative-sequence current with the connecting grid. Zero-sequence voltage for the star and zero-sequence current for the delta configuration are utilized to guarantee dc-capacitor voltage balancing. A general solution for the required zero-sequence voltage/current for any kind of unbalanced condition is provided. The obtained solutions show that a singularity for both configurations exists under specific conditions, leading to infinite zero-sequence requirement. In case of star configuration, this singularity occurs when the amplitude of the positive- and negative-sequence components of the current output of the converter are equal. Analogously, the singularity for the delta configuration occurs when the amplitude of the positive- and negative-sequence components of the voltage at the converter terminals are equal. This singularity impacts the ratings of the system and imposes a limitation in the utilization of the compensator for unbalance compensation purposes, both in industrial and utility applications. Experimental results are presented to validate the theoretical analysis.

E 8 instantons on type-A ALE spaces and supersymmetric field theories

Abstract: We consider the 6d superconformal field theory realized on M5-branes probing the E 8 end-of-the-world brane on the deformed and resolved ℂ 2/ℤ k singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the T 2 compactification, and the star-shaped quiver obtained as the mirror of the T 3 compactification.

From black holes to white holes: a quantum gravitational, symmetric bounce

Abstract: Recently a consistent non-perturbative quantization of the Schwarzschild interior resulting in a bounce from black hole to white hole geometry has been obtained by loop quantizing the Kantowski-Sachs vacuum spacetime. As in other spacetimes where the singularity is dominated by the Weyl part of the spacetime curvature, the structure of the singularity is highly anisotropic in the Kantowski-Sachs vacuum spacetime. As a result the bounce turns out to be in general asymmetric creating a large mass difference between the parent black hole and the child white hole. In this manuscript, we investigate under what circumstances a symmetric bounce scenario can be constructed in the above quantization. Using the setting of Dirac observables and geometric clocks we obtain a symmetric bounce condition which can be satisfied by a slight modification in the construction of loops over which holonomies are considered in the quantization procedure. These modifications can be viewed as quantization ambiguities, and are demonstrated in three different flavors which all lead to a non-singular black to white hole transition with identical masses. Our results show that quantization ambiguities can mitigate or even qualitatively change some key features of physics of singularity resolution. Further, these results are potentially helpful in motivating and constructing symmetric black to white hole transition scenarios.

A tauberian theorem for the conformal bootstrap

Abstract: For expansions in one-dimensional conformal blocks, we provide a rigorous link between the asymptotics of the spectral density of exchanged primaries and the leading singularity in the crossed channel. Our result has a direct application to systems of SL(2, ℝ)-invariant correlators (also known as 1d CFTs). It also puts on solid ground a part of the lightcone bootstrap analysis of the spectrum of operators of high spin and bounded twist in CFTs in d > 2. In addition, a similar argument controls the spectral density asymptotics in large N gauge theories.

Dynamical Symmetry Breaking and Phase Transitions in Driven Diffusive Systems.

Abstract: We study the probability distribution of a current flowing through a diffusive system connected to a pair of reservoirs at its two ends. Sufficient conditions for the occurrence of a host of possible phase transitions both in and out of equilibrium are derived. These transitions manifest themselves as singularities in the large deviation function, resulting in enhanced current fluctuations. Microscopic models which implement each of the scenarios are presented, with possible experimental realizations. Depending on the model, the singularity is associated either with a particle-hole symmetry breaking, which leads to a continuous transition, or in the absence of the symmetry with a first-order phase transition. An exact Landau theory which captures the different singular behaviors is derived.

Three dimensional canonical singularity and five dimensional $ \mathcal{N} $ = 1 SCFT

Abstract: We conjecture that every three dimensional canonical singularity defines a five dimensional $$ \mathcal{N} $$ = 1 SCFT. Flavor symmetry can be found from singularity structure: non-abelian flavor symmetry is read from the singularity type over one dimensional singular locus. The dimension of Coulomb branch is given by the number of compact crepant divisors from a crepant resolution of singularity. The detailed structure of Coulomb branch is described as follows: a) a chamber of Coulomb branch is described by a crepant resolution, and this chamber is given by its Nef cone and the prepotential is computed from triple intersection numbers; b) Crepant resolution is not unique and different resolutions are related by flops; Nef cones from crepant resolutions form a fan which is claimed to be the full Coulomb branch.

Three dimensional canonical singularity and five dimensional N=1 SCFT

Abstract: We conjecture that every three dimensional canonical singularity defines a five dimensional N=1 SCFT. Flavor symmetry can be found from singularity structure: non-abelian flavor symmetry is read from the singularity type over one dimensional singular locus. The dimension of Coulomb branch is given by the number of compact crepant divisors from a crepant resolution of singularity. The detailed structure of Coulomb branch is described as follows: a): A chamber of Coulomb branch is described by a crepant resolution, and this chamber is given by its Nef cone and the prepotential is computed from triple intersection numbers; b): Crepant resolution is not unique and different resolutions are related by flops; Nef cones from crepant resolutions form a fan which is claimed to be the full Coulomb branch.

Starobinsky cosmological model in Palatini formalism

Abstract: We classify singularities in FRW cosmologies, which dynamics can be reduced to the dynamical system of the Newtonian type. This classification is performed in terms of the geometry of a potential function if it has poles. At the sewn singularity, which is of a finite scale factor type, the singularity in the past meets the singularity in the future. We show that such singularities appear in the Starobinsky model in $$f({\hat{R}})={\hat{R}}+\gamma {\hat{R}}^2$$ in the Palatini formalism, when dynamics is determined by the corresponding piecewise-smooth dynamical system. As an effect we obtain a degenerate singularity. Analytical calculations are given for the cosmological model with matter and the cosmological constant. The dynamics of model is also studied using dynamical system methods. From the phase portraits we find generic evolutionary scenarios of the evolution of the universe. For this model, the best fit value of $$\Omega _\gamma =3\gamma H_0^2$$ is equal $$9.70\times 10^{-11}$$ . We consider a model in both Jordan and Einstein frames. We show that after transition to the Einstein frame we obtain both the form of the potential of the scalar field and the decaying Lambda term.

Generalized Wilson-Fisher Critical Points from the Conformal Operator Product Expansion

Abstract: We study possible smooth deformations of the generalized free conformal field theory in arbitrary dimensions by exploiting the singularity structure of the conformal blocks dictated by the null states. We derive in this way, at the first nontrivial order in the e expansion, the anomalous dimensions of an infinite class of scalar local operators, without using the equations of motion. In the cases where other computational methods apply, the results agree.

Weak null singularities in general relativity

Abstract: We construct a class of spacetimes (without symmetry assumptions) satisfying the vacuum Einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2-sphere. The metric of these spacetimes extends continuously beyond the singularities while the Christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries. The construction shows moreover that the singularities are stable in a suitable sense. These singularities are stronger than the impulsive gravitational spacetimes considered by Luk-Rodnianski and are conjecturally the generic type of singularity in the interior of black holes arising from gravitational collapse.

A Cahn‐Hilliard–type equation with application to tumor growth dynamics

Abstract: We consider a Cahn-Hilliard–type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.

Leading Singularities and Classical Gravitational Scattering

Abstract: In this work we propose to use leading singularities to obtain the classical pieces of amplitudes of two massive particles whose only interaction is gravitational Leading singularities are generalizations of unitarity cuts At one-loop we find that leading singularities obtained by multiple discontinuities in the t-channel contain all the classical information As the main example, we show how to obtain a compact formula for the fully relativistic classical one-loop contribution to the scattering of two particles with different masses The non-relativistic limit of the leading singularity agrees with known results in the post-Newtonian expansion We also compute a variety of higher loop leading singularities including some all-loop families and study some of their properties

On irregular singularity wave functions and superconformal indices

Abstract: We generalize, in a manifestly Weyl-invariant way, our previous expressions for irregular singularity wave functions in two-dimensional SU(2) q-deformed Yang-Mills theory to SU(N). As an application, we give closed-form expressions for the Schur indices of all (A N − 1 , A N (n − 1)−1) Argyres-Douglas (AD) superconformal field theories (SCFTs), thus completing the computation of these quantities for the (A N , A M ) SCFTs. With minimal effort, our wave functions also give new Schur indices of various infinite sets of “Type IV” AD theories. We explore the discrete symmetries of these indices and also show how highly intricate renormalization group (RG) flows from isolated theories and conformal manifolds in the ultraviolet to isolated theories and (products of) conformal manifolds in the infrared are encoded in these indices. We compare our flows with dimensionally reduced flows via a simple “monopole vev RG” formalism. Finally, since our expressions are given in terms of concise Lie algebra data, we speculate on extensions of our results that might be useful for probing the existence of hypothetical SCFTs based on other Lie algebras. We conclude with a discussion of some open problems.

The Cosmological Singularity

Abstract: The long story of the oscillatory approach to the initial cosmological singularity and its more recent incarnation in multidimensional universe models is told.

Non-singular black holes and the Limiting Curvature Mechanism: A Hamiltonian perspective

Abstract: We revisit the non-singular black hole solution in (extended) mimetic gravity with a limiting curvature from a Hamiltonian point of view. We introduce a parameterization of the phase space which allows us to describe fully the Hamiltonian structure of the theory. We write down the equations of motion that we solve in the regime deep inside the black hole, and we recover that the black hole has no singularity, due to the limiting curvature mechanism. Then, we study the relation between such black holes and effective polymer black holes which have been introduced in the context of loop quantum gravity. As expected, contrary to what happens in the cosmological sector, mimetic gravity with a limiting curvature fails to reproduce the usual effective dynamics of spherically symmetric loop quantum gravity which are generically not covariant. Nonetheless, we exhibit a theory in the class of extended mimetic gravity whose dynamics reproduces the general shape of the effective corrections of spherically symmetric polymer models, but in an undeformed covariant manner. These covariant effective corrections are found to be always metric dependent, i.e. within the $\bar{\mu}$-scheme, underlying the importance of this ingredient for inhomogeneous polymer models. In that respect, extended mimetic gravity can be viewed as an effective covariant theory which naturally implements a covariant notion of point wise holonomy-like corrections. The difference between the mimetic and polymer Hamiltonian formulations provides us with a guide to understand the deformation of covariance in inhomogeneous polymer models.

Effective loop quantum geometry of Schwarzschild interior

Abstract: The success of loop quantum cosmology to resolve classical singularities of homogeneous models has led to its application to the classical Schwarszchild black hole interior, which takes the form of a homogeneous Kantowski-Sachs model. The first steps of this were done in pure quantum mechanical terms, hinting at the traversable character of the would-be classical singularity, and then others were performed using effective heuristic models capturing quantum effects that allowed a geometrical description closer to the classical one but avoided its singularity. However, the problem of establishing the link between the quantum and effective descriptions was left open. In this work, we propose to fill in this gap by considering the path-integral approach to the loop quantization of the Kantowski-Sachs model corresponding to the Schwarzschild black hole interior. We show that the transition amplitude can be expressed as a path integration over the imaginary exponential of an effective action which just coincides, under some simplifying assumptions, with the heuristic one. Additionally, we further explore the consequences of the effective dynamics. We prove first that such dynamics imply some rather simple bounds for phase-space variables, and in turn---remarkably, in an analytical way---they imply that various phase-space functions that were singular in the classical model are now well behaved. In particular, the expansion rate, its time derivative, and the shear become bounded, and hence the Raychaudhuri equation is finite term by term, thus resolving the singularities of classical geodesic congruences. Moreover, all effective scalar polynomial invariants turn out to be bounded.

On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler Equations

Abstract: In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blowup from smooth data.

Big bounce with finite-time singularity: The F(R) gravity description

Abstract: An alternative to the Big Bang cosmologies is obtained by the Big Bounce cosmologies. In this paper, we study a bounce cosmology with a Type IV singularity occurring at the bouncing point in the context of F(R) modified gravity. We investigate the evolution of the Hubble radius and we examine the issue of primordial cosmological perturbations in detail. As we demonstrate, for the singular bounce, the primordial perturbations originating from the cosmological era near the bounce do not produce a scale-invariant spectrum and also the short wavelength modes after these exit the horizon, do not freeze, but grow linearly with time. After presenting the cosmological perturbations study, we discuss the viability of the singular bounce model, and our results indicate that the singular bounce must be combined with another cosmological scenario, or should be modified appropriately, in order that it leads to a viable cosmology. The study of the slow-roll parameters leads to the same result indicating that the singular bounce theory is unstable at the singularity point for certain values of the parameters. We also conformally transform the Jordan frame singular bounce, and as we demonstrate, the Einstein frame metric leads to a Big Rip singularity. Therefore, the Type IV singularity in the Jordan frame becomes a Big Rip singularity in the Einstein frame. Finally, we briefly study a generalized singular cosmological model, which contains two Type IV singularities, with quite appealing features.

Causality theory for closed cone structures with applications

Abstract: We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including causal ladder, Fermat's principle, notable singularity theorems in their causal formulation, Avez-Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, $K$-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz-Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula a la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity.