Singular Geometry Research Articles

Association schemes based on the subspaces of type [formula omitted] in singular symplectic space over finite fields

The paper provides the construction of association scheme on the subspaces of type (m,s,0) in singular symplectic geometry over finite fields. All intersection numbers of the scheme are computed.

Yamabe flow on manifolds with edges

Let be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schauder‐type estimates for the heat operator on certain Hölder spaces adapted to the singular edge geometry. We apply these estimates to obtain local existence for a variety of quasilinear equations, including the Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe problem using flow techniques in the singular setting.

Construction of Multi-receiver Multi-fold Authentication Codes from Singular Symplectic Geometry over Finite Fields

As an extension of the basic model of MRA-codes, multi-receiver multi-fold authentication codes can use a single key distribution phase for multiple message transmission. A multi-receiver multi-fold authentication code is constructed from the singular symplectic geometry over finite fields in this paper. The parameters and probabilities of success in impersonation and substitution attacks by malicious groups of receivers of this code are computed.

Image denoising method based on dictionary learning with union of two orthonormal bases

Overcomplete dictionary was used to represent an image sparsely in order to improve image denoising performance.The sparse representation may represent efficiently the singular geometry of the images with the redundancy of over-complete dictionary.Global image prior model based on the sparse representation of image patches was presented in Bayesian framework.Then maximum a posteriori probability estimator for denoising image was constructed.The dictionary was composed of two orthonormal bases.A method based on singular value decomposition was used for dictionary learning.The orthonormal property was used to update the one chosen basis effectively.The method can improve the performance of image denoising.The experimental results verify the validity of the method.

Continuous formulation of the loop quantum gravity phase space

In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of general relativity. In particular, we prove an isomorphism between the loop gravity discrete phase space and the symplectic reduction of the continuous phase space with respect to a flatness constraint. This gives for the first time a precise relationship between the continuum and holonomy-flux variables. In our construction, the fluxes not only depend on the three-geometry, but also explicitly on the connection, providing a natural explanation of their non-commutativity. It also clearly shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. This allows us to resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry and the spin foam interpretation in terms of piecewise flat geometry, since we establish that both geometries belong to the same equivalence class. This finally gives us a clear understanding of the relationship between the piecewise flat spin foam geometries and Regge geometries, which are only piecewise-linear flat. The Regge geometry corresponds to metrics whose curvature is concentrated around straight edges, while the loop gravity geometry corresponds to metrics whose curvature is concentrated around not necessarily straight edges.

Profiled spectral lines generated in the field of Kerr superspinars

String Theory suggests existence of primordial Kerr superspinars, extremely compact objects with external spacetime described by the Kerr naked singularity geometry. The primordial Kerr superspinars have to be converted to a near-extreme black hole due to accretion, but they could survive to the era of highly redshifted quasars. We study the shape of the profiled spectral lines generated by radiating rings or the innermost parts of Keplerian discs orbiting the Kerr superspinars. Influence of the superspinar surface location on the profiled lines is also considered. We demonstrate strong difference of the character of the profiled lines generated by radiating rings for all values of the superspinar spin and all values of the inclination angles of the observer when compared to those generated in the field of Kerr black holes. For small and mediate inclination angles there are large quantitative differences in the extension and position of the lines. For large inclination angles even strong qualitative difference appears as the profiled lines have a clear doubled character. The smaller, redshifted region of the profiled line is related to the photons reaching the regions near the superspinar surface. Strong differences are obtained also for profiled lines generated by the innermost parts of Keplerian discs especially in the shape of the line. The influence of the superspinar surface location is reflected in the intermediate parts of the the profiled lines. The line profiles can give a clear signature of the presence of a Kerr superspinar and in principle enable estimates of its surface location since the signatures of the superspinar surface location are of different character as those corresponding to the presence of the black hole horizon.

ON A SINGULAR VARIETY ASSOCIATED TO A POLYNOMIAL MAPPING

In the paper "Geometry of polynomial mapping at infinity via intersection homology" the second and third authors associated to a given polynomial mapping $F : \C^2 \to \C^2$ with nonvanishing jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of the mapping. We generalize this result.

Acceleration of particles and shells by Reissner-Nordström naked singularities

We explore the Reissner-Nordstr\"om naked singularities with a charge $Q$ larger than its mass $M$ from the perspective of particle acceleration. We first consider a collision between two test particles following the radial geodesics in the Reissner-Nordstr\"om naked singular geometry. An initially radially ingoing particle turns back due to the repulsive effect of gravity in the vicinity of a naked singularity. Such a particle then collides with another radially ingoing particle. We show that the center of mass energy of collision taking place at $r\ensuremath{\approx}M$ is unbound, in the limit where the charge transcends the mass by an arbitrarily small amount $0<1\ensuremath{-}M/Q\ensuremath{\ll}1$. The acceleration process we describe avoids fine-tuning of the parameters of the particle geodesics for the unbound center of mass energy of collisions, and the proper time required for the process is also finite. We show that the coordinate time as observed by the distant observer required for the trans-Planckian collisions to occur around the naked singularity with one solar mass is merely of the order of million years, which is much smaller than the Hubble time. On the contrary, the time scale for collisions associated with an extremal black hole in an analogous situation is many orders of magnitude larger than the age of the Universe. We then study the collision of the neutral spherically symmetric shells made up of dust particles. In this case, it is possible to treat the situation by exactly taking into account the gravity due to the shells using Israel's thin shell formalism, and thus this treatment allows us to go beyond the test particle approximation. The center of mass energy of collision of the shells is then calculated in a situation analogous to the test particle case and is shown to be bounded above. However, we find that the energy of a collision between two constituent particles of the shells at the center of mass frame can exceed the Planck energy.

Association schemes based on singular symplectic geometry over finite fields and its application

The paper provides the construction of association scheme on the subspaces of type(ν+k,0,k) in singular symplectic geometry over finite fields. All intersection numbers of the scheme are computed. At last, an authentication code with perfect secrecy from the association scheme is construction.

Spacetimes with singularities

Abstract We report on some advances made in the problem of singularities in general relativity.First is introduced the singular semi-Riemannian geometry for metrics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein's equation, which works for degenerate metric too.Once we make the singularities manageable from mathematical viewpoint, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions. Then we find space-like foliations for these regions, with the implication that the initial data can be preserved in reasonable situations. We propose qualitative models of non-primordial and/or evaporating black holes.We supplement the material with a brief note reporting on progress made since this talk was given, which shows that we can analytically extend the Schwarzschild and Reissner-Nordström metrics at and beyond the singularities, and the singularities can be made degenerate and handled with the mathematical apparatus we developed.

A Hamilton-Jacobi approach to junction problems and application to traffic flows

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a "junction", that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.

TAUB-NUT CRYSTAL

We consider the Gibbons–Hawking metric for a three-dimensional periodic array of multi-Taub-NUT centers, containing not only centers with a positive NUT charge but also ones with a negative NUT charge. The latter are regarded as representing the asymptotic form of the Atiyah–Hitchin metric. The periodic arrays of Taub-NUT centers have close parallels with ionic crystals, where the Gibbons–Hawking potential plays the role of the Coulomb static potential of the ions, and are similarly classified according to their space groups. After a periodic identification and a Z2 projection, the array is transformed by T-duality to a system of NS5-branes with the SU(2) structure, and a further standard embedding yields, though singular, a half-BPS heterotic 5-brane background with warped compact transverse dimensions. A discussion is given on the possibility of probing the singular geometry by two-dimensional gauge theories.

Study of multi-black-hole and ring-singularity apparent horizons

We study critical black hole separations for the formation of a common apparent horizon in systems of $N$ - black holes in a time symmetric configuration. We study in detail the aligned equal mass cases for $N=2,3,4,5$, and relate them to the unequal mass binary black hole case. We then study the apparent horizon of the time symmetric initial geometry of a ring singularity of different radii. The apparent horizon is used as indicative of the location of the event horizon in an effort to predict a critical ring radius that would generate an event horizon of toroidal topology. We found that a good estimate for this ring critical radius is $20/(3\pi) M$. We briefly discuss the connection of this two cases through a discrete black hole 'necklace' configuration.

Evolution of Kerr superspinars due to accretion counterrotating thin discs

String theory predicts the existence of extremely compact objects spinning faster than Kerr black holes. The spacetime exterior to such superspinars is described by Kerr naked singularity geometry breaking the black-hole limit on the internal angular momentum. We demonstrate that the conversion of Kerr superspinars into a near-extreme black hole due to an accretion counterrotating Keplerian disc is much more effective in comparison with the case of a corotating one since both the accreted rest mass necessary for conversion and the evolution time of conversion are by orders smaller for counterrotating discs. The conversion time of Kerr superspinars is given for several accretion regimes, and it is shown that the self-regulated accretion flow implies fastest evolution to the black-hole state. In the final stages of the conversion, Kerr superspinars can serve as very efficient particle accelerators in the region where the black-hole horizon forms.

On meromorphic parameterizations of real algebraic curves

A singular flat geometry may be canonically assigned to a real algebraic curve Γ; namely, via analytic continuation of unit speed parameterization of the real locus \({\Gamma_\mathbb{R}}\) . Globally, the metric \({\rho=|Q|=|q(z)|dzd\bar{z}}\) is given by the meromorphic quadratic differential Q on Γ induced by the standard complex form dx2 + dy2 on \({\mathbb{C}^2=\{(x,y)\}}\) . By considering basic properties of Q, we show that the condition for local arc length parameterization along \({\Gamma_\mathbb{R}}\) to extend meromorphically to the complex plane is quite restrictive: For curves of degree at most four, only lines, circles and Bernoulli lemniscates have such meromorphic parameterizations.

Some New Constructions of Authentication Codes with Arbitration and Multi‐Receiver from Singular Symplectic Geometry

A new construction of authentication codes with arbitration and multireceiver from singular symplectic geometry over finite fields is given. The parameters are computed. Assuming that the encoding rules are chosen according to a uniform probability distribution, the probabilities of success for different types of deception are also computed.

Appearance of Keplerian discs orbiting Kerr superspinars

We study optical phenomena related to the appearance of Keplerian accretion discs orbiting Kerr superspinars predicted by string theory. The superspinar exterior is described by standard Kerr naked singularity geometry breaking the black hole limit on the internal angular momentum (spin). We construct local photon escape cones for a variety of orbiting sources that enable us to determine the superspinars silhouette in the case of distant observers. We show that the superspinar silhouette depends strongly on the assumed edge where the external Kerr spacetime is joined to the internal spacetime governed by string theory and significantly differs from the black hole silhouette. The appearance of the accretion disc is strongly dependent on the value of the superspinar spin in both their shape and frequency shift profile. Apparent extension of the disc grows significantly with the growing spin, while the frequency shift grows with the descending spin. This behaviour differs substantially from the appearance of discs orbiting black holes enabling thus, at least in principle, to distinguish clearly the Kerr superspinars and black holes. In vicinity of a Kerr superspinar the non-escaped photons have to be separated to those captured by the superspinar and those being trapped in its strong gravitational field leading to self-illumination of the disc that could even influence its structure and cause self-reflection effect of radiation of the disc. The amount of trapped photons grows with descending superspinar spin. We thus can expect significant self-illumination effects in the field of Kerr superspinars with near-extreme spin a ∼ 1.

F-theory GUT vacua on compact Calabi-Yau fourfolds

We present compact three-generation F-theory GUT models meeting in particular the constraints of D3-tadpole cancellation and D-term supersymmetry. To this end we explicitly construct elliptically fibered Calabi-Yau fourfolds as complete intersections in a toric ambient space. Toric methods enable us to control the singular geometry of the SU(5) GUT model. The GUT brane wraps a non-generic del Pezzo surface admitting GUT symmetry breaking via hypercharge flux. It is contractible to a curve and we demonstrate the existence of a consistent decoupling limit. We compute the Euler characteristic of the singular Calabi-Yau fourfold to show that our three-generation flux solutions obtained via the spectral cover construction are consistent with D3-tadpole cancellation.

The gravitational dynamics of warped throats

We investigate the time evolution due to gravitational dynamics of a particular spacetime commonly used in brane-cosmology and string compactifications, namely the Klebanov-Strassler geometry, which is achieved by adding a perturbation to the momentum of the static solution. We observe the effects this has on the spacetime and look for evidence of black hole formation or collapsing cycles which could lead to singular geometry. The cycles are seen to commonly re-expand after reaching a minimum value, showing the stability of the solution against perturbations which would change its size. However black holes are observed to form for certain perturbations, which could impede common uses of the throat's stable tip.

Extrinsic curvature of semiconvex subspaces in Alexandrov geometry

In Alexandrov spaces of curvature bounded either above (CBA) or below (CBB), we obtain extrinsic curvature bounds on subspaces associated with semiconcave functions. These subspaces play the role in singular geometry of submanifolds in Riemannian geometry, and arise naturally in many different places. For CBA spaces, we obtain new intrinsic curvature bounds on subspaces. For CBB spaces whose boundary is extrinsically curved, we strengthen Perelman’s concavity theorem for distance from the boundary, deriving corollaries on sharp diameter bounds, contractibility, and rigidity.