Showing papers on "Gravitational singularity published in 2016"
TL;DR: The first family of horizonless supergravity solutions that have the same mass, charges, and angular momenta as general supersymmetric rotating D1-D5-P black holes in five dimensions are constructed.
Abstract: We construct the first family of horizonless supergravity solutions that have the same mass, charges, and angular momenta as general supersymmetric rotating D1-D5-P black holes in five dimensions. This family includes solutions with arbitrarily small angular momenta, deep within the regime of quantum numbers and couplings for which a large classical black hole exists. These geometries are well approximated by the black-hole solution, and in particular exhibit the same near-horizon throat. Deep in this throat, the black-hole singularity is resolved into a smooth cap. We also identify the holographically dual states in the N=(4,4) D1-D5 orbifold conformal field theory (CFT). Our solutions are among the states counted by the CFT elliptic genus, and provide examples of smooth microstate geometries within the ensemble of supersymmetric black-hole microstates.
217 citations
TL;DR: In this paper, a quantum theory for the Schwarzschild interior region of a white-hole spacetime is presented. But it is not a quantum model for the singularity and its effective dynamics possesses a bounce into an expanding regime.
Abstract: The loop quantization of the Schwarzschild interior region, as described by a homogeneous anisotropic Kantowski–Sachs model, is re-examined. As several studies of different—inequivalent—loop quantizations have shown, to date there exists no fully satisfactory quantum theory for this model. This fact poses challenges to the validity of some scenarios to address the black hole information problem. Here we put forward a novel viewpoint to construct the quantum theory that builds from some of the models available in the literature. The final picture is a quantum theory that is both independent of any auxiliary structure and possesses a correct low curvature limit. It represents a subtle but non-trivial modification of the original prescription given by Ashtekar and Bojowald. It is shown that the quantum gravitational constraint is well defined past the singularity and that its effective dynamics possesses a bounce into an expanding regime. The classical singularity is avoided, and a semiclassical spacetime satisfying vacuum Einstein's equations is recovered on the 'other side' of the bounce. We argue that such a metric represents the interior region of a white-hole spacetime, but for which the corresponding 'white hole mass' differs from the original black hole mass. Furthermore, we find that the value of the white hole mass is proportional to the third power of the starting black hole mass.
170 citations
TL;DR: In this paper, the authors studied the information loss from black hole physics in AdS3, focusing on two sharp signatures infecting CFT2 correlators at large central charge c: forbidden singularities arising from Euclidean-time periodicity due to the effective Hawking temperature, and late time exponential decay in the Lorentzian region.
Abstract: We discuss information loss from black hole physics in AdS3, focusing on two sharp signatures infecting CFT2 correlators at large central charge c: ‘forbidden singularities’ arising from Euclidean-time periodicity due to the effective Hawking temperature, and late-time exponential decay in the Lorentzian region. We study an infinite class of examples where forbidden singularities can be resolved by non-perturbative effects at finite c, and we show that the resolution has certain universal features that also apply in the general case. Analytically continuing to the Lorentzian regime, we find that the non-perturbative effects that resolve forbidden singularities qualitatively change the behavior of correlators at times t ∼ S
BH
, the black hole entropy. This may resolve the exponential decay of correlators at late times in black hole backgrounds. By Borel resumming the 1/c expansion of exact examples, we explicitly identify ‘information-restoring’ effects from heavy states that should correspond to classical solutions in AdS3. Our results suggest a line of inquiry towards a more precise formulation of the gravitational path integral in AdS3.
166 citations
TL;DR: In the cosmological scenario in $f(T)$ gravity, this paper derived exact solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe.
Abstract: In the cosmological scenario in $f(T)$ gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Finally, we discuss when a nonlinear $f(T)$-gravity theory provides solutions for the teleparallel equivalence of general relativity and derive conditions for exact solutions of general relativity to solve the field equations of an $f(T)$ theory.
162 citations
TL;DR: In this paper, the authors used the effective field theory (EFT) framework to calculate the tail effect in gravitational radiation reaction, which entered at the fourth post-Newtonian order in the dynamics of a binary system.
Abstract: We use the effective field theory (EFT) framework to calculate the tail effect in gravitational radiation reaction, which enters at the fourth post-Newtonian order in the dynamics of a binary system. The computation entails a subtle interplay between the near (or potential) and far (or radiation) zones. In particular, we find that the tail contribution to the effective action is nonlocal in time and features both a dissipative and a “conservative” term. The latter includes a logarithmic ultraviolet (UV) divergence, which we show cancels against an infrared (IR) singularity found in the (conservative) near zone. The origin of this behavior in the long-distance EFT is due to the point-particle limit—shrinking the binary to a point—which transforms a would-be infrared singularity into an ultraviolet divergence. This is a common occurrence in an EFT approach, which furthermore allows us to use renormalization group (RG) techniques to resum the resulting logarithmic contributions. We then derive the RG evolution for the binding potential and total mass/energy, and find agreement with the results obtained imposing the conservation of the (pseudo) stress-energy tensor in the radiation theory. While the calculation of the leading tail contribution to the effective action involves only one diagram, five are needed for the one-point function. This suggests logarithmic corrections may be easier to incorporate in this fashion. We conclude with a few remarks on the nature of these IR/UV singularities, the (lack of) ambiguities recently discussed in the literature, and the completeness of the analytic post-Newtonian framework.
124 citations
TL;DR: In this article, the evolution of holographic complexity in various AdS/CFT models containing cosmological singularities was studied, and it was shown that a notion of complexity measured by extremal bulk volumes tends to decrease as the singularity is approached in CFT time.
Abstract: We study the evolution of holographic complexity in various AdS/CFT models containing cosmological crunch singularities. We find that a notion of complexity measured by extremal bulk volumes tends to decrease as the singularity is approached in CFT time, suggesting that the corresponding quantum states have simpler entanglement structure at the singularity.
113 citations
TL;DR: In this article, the stability of spatially flat FRW solutions which are geodesically complete is studied, i.e. for which one can follow null (graviton) geodesics both in the past and in the future without ever encountering singularities.
Abstract: We study the stability of spatially flat FRW solutions which are geodesically complete, i.e. for which one can follow null (graviton) geodesics both in the past and in the future without ever encountering singularities. This is the case of NEC-violating cosmologies such as smooth bounces or solutions which approach Minkowski in the past. We study the EFT of linear perturbations around a solution of this kind, including the possibility of multiple fields and fluids. One generally faces a gradient instability which can be avoided only if the operator (3) R δ N is present and its coefficient changes sign along the evolution. This operator (typical of beyond-Horndeski theories) does not lead to extra degrees of freedom, but cannot arise starting from any theory with second-order equations of motion. The change of sign of this operator prevents to set it to zero with a generalised disformal transformation.
112 citations
TL;DR: In this article, the authors studied the singularity correspondence between the Jordan and Einstein frames for various F(R) gravity theories, in the absence of any matter fluids, and showed that it is possible to have various correspondences of finite time singularities, and that a singular cosmology in one frame might be non-singular in another frame.
111 citations
TL;DR: In this article, the duality between ALE singularities in M-theory and 7-branes on a circle in Ftheory was revisited and it was shown that a frozen ALE singularity maps to a circle compactification involving a rotation of the plane transverse to the 7brane, showing an interesting correspondence between commuting triples in simply-laced groups.
Abstract: We revisit the duality between ALE singularities in M-theory and 7-branes on a circle in F-theory. We see that a frozen M-theory singularity maps to a circle compactification involving a rotation of the plane transverse to the 7-brane, showing an interesting correspondence between commuting triples in simply-laced groups and Kodaira’s classification of singular elliptic fibrations. Our analysis strongly suggests that the O7+ plane is the only completely frozen F-theory singularity.
83 citations
TL;DR: In this paper, the authors present a systematic analysis of homogeneous and isotropic cosmologies in a particular Horndeski model with Galileon shift symmetry, containing also a Λ-term and a matter.
Abstract: We present a systematic analysis of homogeneous and isotropic cosmologies in a particular Horndeski model with Galileon shift symmetry, containing also a Λ-term and a matter. The model, sometimes called Fab Five, admits a rich spectrum of solutions. Some of them describe the standard late time cosmological dynamic dominated by the Λ-term and matter, while at the early times the universe expands with a constant Hubble rate determined by the value of the scalar kinetic coupling. For other solutions the Λ-term and matter are screened at all times but there are nevertheless the early and late accelerating phases. The model also admits bounces, as well as peculiar solutions describing ``the emergence of time''. Most of these solutions contain ghosts in the scalar and tensor sectors. However, a careful analysis reveals three different branches of ghost-free solutions, all showing a late time acceleration phase. We analyse the dynamical stability of these solutions and find that all of them are stable in the future, since all their perturbations stay bounded at late times. However, they all turn out to be unstable in the past, as their perturbations grow violently when one approaches the initial spacetime singularity. We therefore conclude that the model has no viable solutions describing the whole of the cosmological history, although it may describe the current acceleration phase. We also check that the flat space solution is ghost-free in the model, but it may acquire ghost in more general versions of the Horndeski theory.
82 citations
TL;DR: In this paper, a simple modification of the longitudinal mode in General Relativity is proposed, which incorporates the idea of limiting curvature, so that the singularities in contracting Friedmann and Kasner universes are avoided and instead, the universe has a regular bounce which takes place during the time inversely proportional to the square root of the limiting curvatures.
Abstract: We find a simple modification of the longitudinal mode in General Relativity which incorporates the idea of limiting curvature. In this case the singularities in contracting Friedmann and Kasner universes are avoided, and instead, the universe has a regular bounce which takes place during the time inversely proportional to the square root of the limiting curvature. Away from the bounce, corrections to General Relativity are negligible. In addition the non-singluar modification of General Relativity delivers for free a realistic candidate for Dark Matter.
TL;DR: In this paper, it was shown that the relative singularity category Δ R ( A ) of A has a number of pleasant properties, such as being hom-finite, and that it determines the classical singularity categories D s g ( R ) of Buchweitz and Orlov as a certain canonical quotient category.
TL;DR: In this article, the authors investigated the qualitative features of binary black hole shadows using the model of two extremally charged black holes in static equilibrium (a Majumdar-Papapetrou solution) and developed an appropriate symbolic dynamics to describe planar null geodesics on the double black hole spacetime.
Abstract: We investigate the qualitative features of binary black hole shadows using the model of two extremally charged black holes in static equilibrium (a Majumdar–Papapetrou solution). Our perspective is that binary spacetimes are natural exemplars of chaotic scattering, because they admit more than one fundamental null orbit, and thus an uncountably infinite set of perpetual null orbits which generate scattering singularities in initial data. Inspired by the three-disc model, we develop an appropriate symbolic dynamics to describe planar null geodesics on the double black hole spacetime. We show that a one-dimensional (1D) black hole shadow may be constructed through an iterative procedure akin to the construction of the Cantor set; thus the 1D shadow is self-similar. Next, we study non-planar rays, to understand how angular momentum affects the existence and properties of the fundamental null orbits. Taking slices through 2D shadows, we observe three types of 1D shadow: regular, Cantor-like, and highly chaotic. The switch from Cantor-like to regular occurs where outer fundamental orbits are forbidden by angular momentum. The highly chaotic part is associated with an unexpected feature: stable and bounded null orbits, which exist around two black holes of equal mass M separated by , where . To show how this possibility arises, we define a certain potential function and classify its stationary points. We conjecture that the highly chaotic parts of the 2D shadow possess the Wada property. Finally, we consider the possibility of following null geodesics through event horizons, and chaos in the maximally extended spacetime.
TL;DR: In this article, the authors study the rainbow deformation of the FRW cosmology in both Einstein gravity and Gauss-Bonnet gravity and demonstrate that the singularity can be removed by using a rainbow function motivated from the hard spectra of gamma-ray bursts.
Abstract: In this paper, we will study the rainbow deformation of the FRW cosmology in both Einstein gravity and Gauss-Bonnet gravity. We will demonstrate that the singularity in the FRW cosmology can be removed because of the rainbow deformation of the FRW metric. We will obtain the general constraints required for the FRW cosmology to be free from singularities. It will be observed that the inclusion of Gauss-Bonnet gravity can significantly change the constraints required to obtain a nonsingular universes. We will use a rainbow functions motivated from the hard spectra of gamma-ray bursts to deform the FRW cosmology, and it will be explicitly demonstrated that such a deformation removes the singularity in the FRW cosmology.
TL;DR: In this paper, a criterion for the existence of a metric of curvature 1 on a 2-sphere with n conical singularities of prescribed angles 2πϑ 1,...,2π ϑn and non-coaxial holonomy is given.
Abstract: In this article we give a criterion for the existence of a metric of curvature 1 on a 2-sphere with n conical singularities of prescribed angles 2πϑ1,...,2πϑn and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in ϑ1,...,ϑn.
TL;DR: For a class of nonspherical-wrist manipulators, i.e., redundant or nonredundant, unified singularity analysis and computation-effective avoidance methods are proposed and theoretical analysis shows that the computation costs of the reduced-order approaches are only 1/3–1/2 of the traditional methods.
Abstract: For a class of nonspherical-wrist manipulators, i.e., redundant or nonredundant, we propose unified singularity analysis and computation-effective avoidance methods. First, we construct a unified model to describe this class of manipulators and derive the kinematics equation and its modified form. Second, the Jacobian matrix of an $n$ -degree-of-freedom (DOF) manipulator is partitioned into block triangle form, and the singularity conditions are isolated and collected in a $3\times(n-3)$ submatrix. By analyzing the rank degeneracy conditions of the submatrix, singularity configurations are identified. Third, based on the partitioned Jacobian matrix, the kinematics equation is decomposed into two smaller dimension subequations, only one of which (called singular subequation, as determined by the $3\times(n-3)$ submatrix) contains singularities. Finally, reduced-order approaches and the singularity parameter optimization (SPO) method are presented to plan the singularity-free trajectories by handling the singular subequation. Theoretical analysis shows that the computation costs of the reduced-order approaches are only 1/3–1/2 of the traditional methods. The SPO method is even more efficient than the reduced-order methods. Simulation results verify the proposed methods.
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TL;DR: In this article, Borel et al. studied the information loss from black hole physics in AdS$_3, focusing on two sharp signatures infecting CFT$_2$ correlators at large central charge $c$: forbidden singularities arising from Euclidean-time periodicity due to the effective Hawking temperature, and late-time exponential decay in the Lorentzian region.
Abstract: We discuss information loss from black hole physics in AdS$_3$, focusing on two sharp signatures infecting CFT$_2$ correlators at large central charge $c$: 'forbidden singularities' arising from Euclidean-time periodicity due to the effective Hawking temperature, and late-time exponential decay in the Lorentzian region. We study an infinite class of examples where forbidden singularities can be resolved by non-perturbative effects at finite $c$, and we show that the resolution has certain universal features that also apply in the general case. Analytically continuing to the Lorentzian regime, we find that the non-perturbative effects that resolve forbidden singularities qualitatively change the behavior of correlators at times $t \sim S_{BH}$, the black hole entropy. This may resolve the exponential decay of correlators at late times in black hole backgrounds. By Borel resumming the $1/c$ expansion of exact examples, we explicitly identify 'information-restoring' effects from heavy states that should correspond to classical solutions in AdS$_3$. Our results suggest a line of inquiry towards a more precise formulation of the gravitational path integral in AdS$_3$.
TL;DR: In this paper, it was shown that wormholes generated by spherically symmetric electric fields can resolve black hole singularities without necessarily removing curvature divergences, by studying geodesic completeness, the behavior of time-like congruences going through the divergent region, and by scattering of waves off the wormhole.
Abstract: In certain extensions of General Relativity, wormholes generated by spherically symmetric electric fields can resolve black hole singularities without necessarily removing curvature divergences. This is shown by studying geodesic completeness, the behavior of time-like congruences going through the divergent region, and by means of scattering of waves off the wormhole. This provides an example of the logical independence between curvature divergences and space-time singularities, concepts very often identified with each other in the literature.
TL;DR: Numerical studies show that the proposed model is the right one for image restorations, when the underlying solutions are piecewise smooth, and it is proved rigorously that the discrete model converges to the variational model as image resolution goes to infinity.
TL;DR: In this paper, a set of geometries that regularize the classical singular behavior and present modifications of the near-horizon Schwarzschild geometry are proposed, as a result of the propagation of non-perturbative ultraviolet effects originated in regions of high curvature.
Abstract: The gravitational collapse of massive stars serves to manifest the most severe deviations of general relativity with respect to Newtonian gravity: the formation of horizons and spacetime singularities. Both features have proven to be catalysts of deep physical developments, especially when combined with the principles of quantum mechanics. Nonetheless, it is seldom remarked that it is hardly possible to combine all these developments into a unified theoretical model, while maintaining reasonable prospects for the independent experimental corroboration of its different parts. In this paper we review the current theoretical understanding of the physics of gravitational collapse in order to highlight this tension, stating the position that the standard view on evaporating black holes stands for. This serves as the motivation for the discussion of a recent proposal that offers the opposite perspective, represented by a set of geometries that regularize the classical singular behavior and present modifications of the near-horizon Schwarzschild geometry as the result of the propagation of non-perturbative ultraviolet effects originated in regions of high curvature. We present an extensive exploration of the necessary steps on the explicit construction of these geometries, and discuss how this proposal could change our present understanding of astrophysical black holes and even offer the possibility of detecting genuine ultraviolet effects in gravitational-wave experiments.
TL;DR: The gravitational anomaly, which appears as a finite size correction on smooth surfaces, dominates geometric transport on singular surfaces and determines the fine structure of the electronic density at the conical point.
Abstract: We study quantum Hall states on surfaces with conical singularities. We show that the electronic fluid at the cone tip possesses an intrinsic angular momentum, which is due solely to the gravitational anomaly. We also show that quantum Hall states behave as conformal primaries near singular points, with a conformal dimension equal to the angular momentum. Finally, we argue that the gravitational anomaly and conformal dimension determine the fine structure of the electronic density at the conical point. The singularities emerge as quasiparticles with spin and exchange statistics arising from adiabatically braiding conical singularities. Thus, the gravitational anomaly, which appears as a finite size correction on smooth surfaces, dominates geometric transport on singular surfaces.
TL;DR: An exact solution of two singularities in the teleparallel equivalent to general relativity theory has been obtained in this paper, where a holographic visualization of the binary black holes (BBHs) space-time, due to the non vanishing torsion scalar field, has been given.
Abstract: An exact solution of two singularities in the teleparallel equivalent to general relativity theory has been obtained. A holographic visualization of the binary black holes (BBHs) space-time, due to the non vanishing torsion scalar field, has been given. The acceleration tensor of BBHs space-time has been calculated. The results identify the repulsive gravity zones of the BBHs field. The total conserved quantities of the BBHs has been evaluated. Possible gravitational radiation emission by the system has been calculated without assuming a weak field initial data.
TL;DR: In this paper, the authors considered the CHY-construction of bi-adjoint ϕ 3 theory and presented the explicit formula for two-loop planar integrands.
Abstract: In this paper, by treating massive loop momenta as massless momenta in higher dimensions, we are able to treat all-loop scattering equations as tree ones. As an application of the new perspective, we consider the CHY-construction of bi-adjoint ϕ
3 theory. We present the explicit formula for two-loop planar integrands. We discuss in details how to subtract various forward singularities in the construction. We count the number of terms obtained by our formula and by direct Feynman diagram calculation and find the perfect match, thus provide a strong support for our results.
TL;DR: In this article, the relation between the Jordan-Einstein frame transition and the possible description of the crossing of singularities in flat Friedmann universes, using the fact that the regular evolution in one frame can correspond to crossing singularities of the other frame.
Abstract: We study the relation between the Jordan-Einstein frame transition and the possible description of the crossing of singularities in flat Friedmann universes, using the fact that the regular evolution in one frame can correspond to crossing singularities in the other frame. We show that some interesting effects arise in simple models such as one with a massless scalar field or another wherein the potential is constant in the Einstein frame. The dynamics in these models and in their conformally coupled counterparts are described in detail, and a method for the continuation of such cosmological evolutions beyond the singularity is developed. We compare our approach with some other, recently developed, approaches to the problem of the crossing of singularities.
TL;DR: In this article, the existence of cosmological singularities and the conditions that guarantee late-time acceleration in the Hu-Sawicki $f(R) model were investigated.
Abstract: Modified gravity has attracted much attention over the last few years and remains a potential candidate for dark energy. In particular, the so-called viable $f(R)$ gravity theories, which are able to both recover general relativity and produce late-time cosmic acceleration, have been widely studied in recent literature. Nevertheless, extended theories of gravity suffer from several shortcomings which compromise their ability to provide realistic alternatives to the standard cosmological $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ Concordance model. We address the existence of cosmological singularities and the conditions that guarantee late-time acceleration, assuming reasonable energy conditions for standard matter in the so-called Hu-Sawicki $f(R)$ model, currently among the most widely studied modifications to general relativity. Then using the supernovae Ia Union 2.1 catalogue, we further constrain the free parameters of this model. The combined analysis of both theoretical and observational constraints sheds some light on the viable parameter space of these models and the form of the underlying effective theory of gravity.
TL;DR: In this article, the authors investigate the qualitative features of binary black hole shadows using the model of two extremally charged black holes in static equilibrium (a Majumdar-Papapetrou solution).
Abstract: We investigate the qualitative features of binary black hole shadows using the model of two extremally charged black holes in static equilibrium (a Majumdar--Papapetrou solution). Our perspective is that binary spacetimes are natural exemplars of chaotic scattering, because they admit more than one fundamental null orbit, and thus an uncountably-infinite set of perpetual null orbits which generate scattering singularities in initial data. Inspired by the three-disc model, we develop an appropriate symbolic dynamics to describe planar null geodesics on the double black hole spacetime. We show that a one-dimensional (1D) black hole shadow may constructed through an iterative procedure akin to the construction of the Cantor set; thus the 1D shadow is self-similar. Next, we study non-planar rays, to understand how angular momentum affects the existence and properties of the fundamental null orbits. Taking slices through 2D shadows, we observe three types of 1D shadow: regular, Cantor-like, and highly chaotic. The switch from Cantor-like to regular occurs where outer fundamental orbits are forbidden by angular momentum. The highly chaotic part is associated with an unexpected feature: stable and bounded null orbits, which exist around two black holes of equal mass $M$ separated by $a_1 < a < \sqrt{2} a_1$, where $a_1 = 4M/\sqrt{27}$. To show how this possibility arises, we define a certain potential function and classify its stationary points. We conjecture that the highly chaotic parts of the 2D shadow possess the Wada property. Finally, we consider the possibility of following null geodesics through event horizons, and chaos in the maximally-extended spacetime.
TL;DR: In this article, a specific Hordenski scalar-gravity mimetic model is investigated within a FLWR space-time, and the mimetic scalar field is implemented via a Lagrangian multiplier.
Abstract: A specific Hordenski scalar-gravity mimetic model is investigated within a FLWR space-time. The mimetic scalar field is implemented via a Lagrangian multiplier, and it is shown that the model has equations of motion formally similar to the original simpler mimetic matter model of Chamseddine–Mukhanov–Vikman. Several exact solutions describing inflation, bounces, and future-time singularities are presented and discussed.
TL;DR: In this paper, the authors deduce some highly nontrivial consequences for holographic quantum gravity, including the following: (i) certain cosmological bounces are forbidden, (ii) generic singularities inside black holes cannot be resolved, and (iii) traversable wormholes do not exist.
Abstract: Two quantum field theories whose Hilbert spaces do not overlap cannot transmit a signal to one another. From this simple principle, we deduce some highly nontrivial consequences for holographic quantum gravity. These include: (i) certain cosmological bounces are forbidden, (ii) generic singularities inside black holes cannot be resolved, and (iii) traversable wormholes do not exist. At the classical level, this principle rules out certain types of naked singularities and suggests that new singularity theorems should exist.
TL;DR: In this paper, the authors consider the case when the zero spectral value is multiple and show that the dimension of the projected state on the center manifold is none other than the sum of the dimensions of the generalized eigenspaces associated with spectral values with zero real parts.
Abstract: The analysis of time-delay systems mainly relies on detecting and understanding the spectral values bifurcations when crossing the imaginary axis. This paper deals with the zero singularity, essentially when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an algebraic multiplicity two and a geometric multiplicity one, known as the Bogdanov-Takens singularity. Moreover, in some cases the codimension of the zero spectral value exceeds the number of the coupled scalar-differential equations. Nevertheless, to the best of the author's knowledge, the bounds of such a multiplicity have not been deeply investigated in the literature. It is worth mentioning that the knowledge of such an information is crucial for nonlinear analysis purposes since the dimension of the projected state on the center manifold is none other than the sum of the dimensions of the generalized eigenspaces associated with spectral values with zero real parts. Motivated by a control-oriented problems, this paper provides an answer to this question for time-delay systems, taking into account the parameters' algebraic constraints that may occur in applications. We emphasize the link between such a problem and the incidence matrices associated with the Birkhoff interpolation problem. In this context, symbolic algorithms for LU-factorization for functional confluent Vandermonde as well as some classes of bivariate functional Birkhoff matrices are also proposed.
TL;DR: In this article, it was shown that for a system with f degrees of freedom, a non-degenerate stationary point with index r causes a discontinuity (for r even) or divergence (r odd) of the ( f − 1 ) ǫth derivative of both density and flow of the spectrum.