Invariant Characterization Research Articles

Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below

After establishing some new global facts (like a measure theoretic structure theorem and a new invariant characterization of Sobolev functions) about complex-valued functions with bounded variation on arbitrary noncompact Riemannian manifolds, we extend results of Miranda/the second author/Paronetto/Preunkert and of Carbonaro/Mauceri on the heat semigroup characterization of the variation of $$\mathsf {L}^1$$ -functions to a class of Riemannian manifolds with possibly unbounded from below Ricci curvature.

Invariant characterization of the Kerr spacetime: Locating the horizon and measuring the mass and spin of rotating black holes using curvature invariants

We provide an invariant characterization of the physical properties of the Kerr spacetime. We introduce two dimensionless invariants, constructed out of some known curvature invariants, that act as detectors for the event horizon and ergosurface of the Kerr black hole. We also show that the mass and angular momentum can be extracted from local measurements of the curvature invariants, which in the weak field limit could be used to approximate the total angular momentum and mass of a system of merging black holes. Finally, we introduce a dimensionless invariant that gives a local measure of the "Kerrness" of the spacetime.

Intrinsic properties of surfaces with singularities

In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R3 are Kossowski metrics, and the pull-back metrics of surfaces consisting only of cross cap singularities are Whitney metrics. Since the singular sets of Kossowski metrics are the union of regular curves on the domains of definitions, and Whitney metrics admit only isolated singularities, these two classes of metrics are disjoint. In this paper, we give several characterizations of intrinsic invariants of cuspidal edges and cross caps in these classes of metrics. Moreover, we prove Gauss–Bonnet type formulas for Kossowski metrics and for Whitney metrics on compact 2-manifolds.

Asymptotics with a positive cosmological constant: I. Basic framework

The asymptotic structure of the gravitational field of isolated systems has been analyzed in great detail in the case when the cosmological constant Λ is zero. The resulting framework lies at the foundation of research in diverse areas in gravitational science. Examples include: (i) positive energy theorems in geometric analysis; (ii) the coordinate invariant characterization of gravitational waves in full, nonlinear general relativity; (iii) computations of the energy–momentum emission in gravitational collapse and binary mergers in numerical relativity and relativistic astrophysics; and (iv) constructions of asymptotic Hilbert spaces to calculate S-matrices and analyze the issue of information loss in the quantum evaporation of black holes. However, by now observations have led to a strong consensus that Λ is positive in our universe. In this paper we show that, unfortunately, the standard framework does not extend from the case to the case in a physically useful manner. In particular, we do not have positive energy theorems, nor an invariant notion of gravitational waves in the nonlinear regime, nor asymptotic Hilbert spaces in dynamical situations of semi-classical gravity. A suitable framework to address these conceptual issues of direct physical importance is developed in subsequent papers.

Invariant characterization of the growing and decaying density modes in LTB dust models

We obtain covariant expressions that generalize the growing and decaying density modes of linear perturbation theory of dust sources by means of the exact density perturbation from the formalism of quasi-local scalars associated to weighted proper volume averages in LTB dust models. The relation between these density modes and theoretical properties of generic LTB models is thoroughly studied by looking at the evolution of the models through a dynamical system whose phase space is parametrized by variables directly related to the modes themselves. The conditions for absence of shell crossings and sign conditions on the modes become interrelated fluid flow preserved constraints that define sub-cases of LTB models as phase space invariant subspaces. In the general case (both density modes being nonzero) the evolution of phase space trajectories exhibits the expected dominance of the decaying/growing in the early/late evolution times defined by past/future attractors characterized by asymptotic density inhomogeneity. In particular, the growing mode is also dominant for collapsing layers that terminate in a future attractor associated with a ‘Big Crunch’ singularity, which is qualitatively different from the past attractor marking the ‘Big Bang’. Suppression of the decaying mode modifies the early time evolution, with phase space trajectories emerging from an Einstein–de Sitter past attractor associated with homogeneous conditions. Suppression of the growing mode modifies the late time evolution as phase space trajectories terminate in future attractors associated with homogeneous states. General results are obtained relating the signs of the density modes and the type of asymptotic density profile (clump or void). A critical review is given of previous attempts in the literature to define these density modes for LTB models.

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

We consider time reversal transformations to obtain twofold orthogonal splittings of any tensor on a Lorentzian space of arbitrary dimension n. Applied to the Weyl tensor of a spacetime, this leads to a definition of its electric and magnetic parts relative to an observer (defined by a unit timelike vector field u), in any dimension. We study the cases where one of these parts vanishes in detail, i.e., purely electric (PE) or magnetic (PM) spacetimes. We generalize several results from four to higher dimensions and discuss new features of higher dimensions. For instance, we prove that the only permitted Weyl types are G, Ii and D, and discuss the possible relation of u with the Weyl aligned null directions (WANDs); we provide invariant conditions that characterize PE/PM spacetimes, such as Bel–Debever-like criteria, or constraints on scalar invariants, and connect the PE/PM parts to the kinematic quantities of u; we present conditions under which direct product spacetimes (and certain warps) are PE/PM, which enables us to construct explicit examples. In particular, it is also shown that all static spacetimes are necessarily PE, while stationary spacetimes (such as spinning black holes) are in general neither PE nor PM. Whereas ample classes of PE spacetimes exist, PM solutions are elusive; specifically, we prove that PM Einstein spacetimes of type D do not exist, in any dimension. Finally, we derive corresponding results for the electric/magnetic parts of the Riemann tensor, which is useful when considering spacetimes with matter fields, and moreover leads to first examples of PM spacetimes in higher dimensions. We also note in passing that PE/PM Weyl (or Riemann) tensors provide examples of minimal tensors, and we make the connection hereof with the recently proved alignment theorem (Hervik 2011 Class. Quantum Grav. 28 215009). This in turn sheds new light on the classification of the Weyl tensors based on null alignment, providing a further invariant characterization that distinguishes the (minimal) types G/I/D from the (non-minimal) types II/III/N.

Invariants of a family of scalar second-order ordinary differential equations

The family of second-order equations with cubic nonlinearity in the first-order derivative is closed with respect to an arbitrary point change of variables. Algebraic and differential invariants for these equations which depend on the first-order derivatives are considered. We solve completely the equivalence problem for this family of equations in the generic case and also for degenerate types of these equations. Fifty equations having the Painlevé property, which have been classified by Painlevé and Gambier, belong to this remarkable family. Algebraic invariants of all these equations are calculated and constant invariants are listed in the paper. Invariant characterization of the fourth Painlevé equation is given.

A set of invariant quality factors measuring the deviation from the Kerr metric

A number of scalar invariant characterizations of the Kerr solution are presented. These characterizations come in the form of {\em quality factors} defined in stationary space-times. A quality factor is a scalar quantity varying in the interval $[0,1]$ with the value 1 being attained if and only if the space-time is locally isometric to the Kerr solution. No knowledge of the Kerr solution is required to compute these quality factors. A number of different possibilities arise depending on whether the space-time is Ricci-flat and asymptotically flat, just Ricci-flat, or Ricci non-flat. In each situation a number of quality factors are constructed and analysed. The relevance of these quality factors is clear in any situation where one seeks a rigorous formulation of the statement that a space-time is "close" to the Kerr solution, such as: its non-linear stability problem, the asymptotic settlement of a radiating isolated system undergoing gravitational collapse, or in the formulation of some uniqueness results.

Invariant theoretic characterization of subdiscriminants of matrices

An invariant theoretic characterization of subdiscriminants of matrices is given. The structure as a module over the special orthogonal group of the minimal degree non-zero homogeneous component of the vanishing ideal of the variety of real symmetric matrices with a bounded number of different eigenvalues is investigated. These results are applied to the study of sum of squares presentations of subdiscriminants of real symmetric matrices.

Invariant classification of vacuum pp-waves

We solve the equivalence problem for vacuum pp-wave spacetimes by employing the Karlhede algorithm. Our main result is a suite of Cartan invariants that allows for the complete invariant classification of the vacuum pp-waves. In particular, we derive the invariant characterization of the G2 and G3 sub-classes in terms of these invariants. It is known [J. M. Collins, “The Karlhede classification of type N vacuum spacetimes,” Class. Quantum Grav. 8, 1859–1869 (1991)10.1088/0264-9381/8/10/011] that the invariant classification of vacuum pp-waves requires at most the fourth order covariant derivative of the curvature tensor, but no specific examples requiring the fourth order were known. Using our comprehensive classification, we prove that the q ⩽ 4 bound is sharp and explicitly describe all such maximal order solutions.

SPACETIMES WITH ALL SCALAR CURVATURE INVARIANTS IN TERMS OF A COSMOLOGICAL CONSTANT

In this paper we provide an invariant characterization for all spacetimes with all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives vanishing except those zeroth-order curvature invariants expressed as polynomials in Λ, the cosmological constant. Using this invariant description we provide an explicit forms for the metric.

Positive invariant set estimation of adaptive controller in the presence of input saturation

SUMMARYFor a class of linear dynamical systems with constant unknown parameters, an adaptive control scheme is developed that provides stable adaptation in the presence of input magnitude constraints. Whereas for open‐loop stable systems the results are global, for open‐loop unstable systems, the problem of nonconservative estimation of the nonempty positive invariant set is cast into an LMI framework, which can be efficiently solved numerically via convex optimization. To achieve this, a standard result toward invariant set characterization is appropriately extended to accommodate bounded disturbance and model uncertainties. In addition to closed‐loop stability, performance bounds of the adaptive closed‐loop system are analyzed, and the degradation due to the possible control deficiency is quantified. Simulation examples of aerospace applications are included to illustrate the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.

Rotating black holes, global symmetry and first order formalism

In this paper we consider axisymmetric black holes in supergravity and address the general issue of defining a first order description for them. The natural setting where to formulate the problem is the De Donder-Weyl-Hamilton-Jacobi theory associated with the effective two-dimensional sigma-model action describing the axisymmetric solutions. We write the general form of the two functions S m defining the first-order equations for the fields. It is invariant under the global symmetry group G (3) of the sigma-model. We also discuss the general properties of the solutions with respect to these global symmetries, showing that they can be encoded in two constant matrices belonging to the Lie algebra of G (3), one being the Nöther matrix of the sigma model, while the other is non-zero only for rotating solutions. These two matrices allow a G (3)-invariant characterization of the rotational properties of the solution and of the extremality condition. We also comment on extremal, under-rotating solutions from this point of view.

Whitney tower concordance of classical links

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link in the 3-sphere. Applications include computation of the grope filtration, and new geometric characterizations of Milnor's link invariants.

A [formula omitted]- and [formula omitted]-invariant characterization of product form and decomposition in stochastic Petri nets

Structural product form and decomposition results for stochastic Petri nets are surveyed, unified and extended. The contribution is threefold. First, the literature on structural results for product form over the number of tokens at the places is surveyed and rephrased completely in terms of T-invariants. Second, based on the underlying concept of group-local-balance, the product form results for stochastic Petri nets are demarcated and an intuitive explanation is provided of these results based on T-invariants, only. Third, a decomposition result is provided that is completely formulated in terms of both T-invariantsandP-invariants.

Minkowskian Isotropic Media and the Perfect Electromagnetic Conductor

The perfect electromagnetic conductor (PEMC) was introduced as an observer-independent “axion medium” that generalizes the concepts of perfect electric conductor (PEC) and perfect magnetic conductor (PMC). Following the original boundary definition, its 3-D medium definition corresponds to a 4-D representation that is, actually, observer-dependent (i.e., it is not isotropic for the whole class of inertial observers), leading to a nonunique characterization of the electromagnetic field inside. This characterization of the PEMC, then, violates the boundary conditions-unless some extraneous waves, called “metafields,” are surgically extracted from the final solution. In this paper, using spacetime algebra, we define the PEMC as the unique limit of the most general class of isotropic media in Minkowskian spacetime, which we call Minkowskian isotropic media (MIM). An MIM is actually a “dilaton-axion medium.” Its isotropy is a Lorentz invariant characterization: It is an observer-independent property, contrary to isotropy in 3-D Gibbsian characterization. Hence, a more natural definition of a PEMC is herein presented: It leads to a unique electromagnetic field in its interior; it corresponds, though, to the same original boundary definition. This new approach is applied to the analysis of an air-MIM interface that, as a particular case, reduces to an air-PEMC interface.

Extending morphological covariance

Mathematical morphology conversely to linear image analysis approaches specialises in capturing the spatial relations among pixels. This inherent potential has been exploited in the context of texture characterisation, with granulometry along with morphological covariance being the two main tools of the morphological arsenal for this task. However, with the advent of new and powerful texture analysis approaches in the last years (e.g.local binary patterns, MR8), they have been left relatively behind the state-of-the-art, in the light of the present challenges of this field, particularly illumination, rotation and scale invariant characterisation. In this paper, we present a set of extensions for morphological covariance, inspired from differential morphological profiles, that enhance its rotation and illumination invariance capacity. The proposed approach is tested extensively against the state-of-the-art, using the Outex, CUReT, KTH-TIPS, KTH-TIPS2 and ALOT databases, where it exhibits either a superior or comparable performance.

On the invariant causal characterization of singularities in spherically symmetric spacetimes

The causal character of singularities is often studied in relation to the existence of naked singularities and the subsequent possible violation of the cosmic censorship conjecture. Generally, one constructs a model in the framework of general relativity described in some specific coordinates and finds an ad hoc procedure to analyze the character of the singularity. In this paper, we show that the causal character of the zero-areal-radius (R = 0) singularity in spherically symmetric models is related to some specific invariants. In this way, if some assumptions are satisfied, one can ascertain the causal character of the singularity algorithmically through the computation of these invariants and, therefore, independent of the coordinates used in the model.

Fubini–Griffiths–Harris Rigidity of Homogeneous Varieties

Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.

Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids

Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.