Invariant Characterization Research Articles

Measuring topological invariants for higher-order exceptional points in quantum three-mode systems

Owing to the presence of exceptional points (EPs), non-Hermitian (NH) systems can display intriguing topological phenomena without Hermitian analogs. However, experimental characterizations of exceptional topological invariants have been restricted to second-order EPs (EP2s) in classical or semiclassical systems. We here propose an NH multi-mode system with higher-order EPs, each of which is underlain by a multifold-degenerate multipartite entangled eigenstate. We implement the NH model by controllably coupling a Josephson-junction-based electronic mode to two microwave resonators. We experimentally quantify the topological invariant for an EP3, by mapping out the complex eigenspectra of the tripartite system along a loop surrounding this EP3 in the parameter space. The nonclassicality of the realized topology is manifested by the observed quantum correlations in the corresponding eigenstates. Our results extend research of exceptional topology to fully quantum-mechanical models with multipartite entangled eigenstates.

Schouten–Codazzi gravity

Abstract We propose a new phenomenological second order gravity theory to be denoted as ''Schouten-Codazzi' Gravity'' (SCG), as it is based on Schouten and Codazzi tensors. The theory is related, but is clearly distinct from Cotton Gravity. By assuming as source the energy momentum of General Relativity, we form a second order system with its geometric sector given by the sum of the Schouten tensor and a generic second order symmetric tensor complying with the following properties: (i) it must satisfy the Codazzi differential condition and (ii) it must be concomitant with the invariant characterization based on the algebraic structure of curvature tensors for specific spacetimes or classes of spacetimes. We derive and briefly discuss the properties of SCG solutions for static spherical symmetry (vacuum and perfect fluid), FLRW models and spherical dust fluids. While we do recognize that SCG is ``work in progress'' in an incipient stage that still requires significant theoretical development, we believe that the theory provides valuable guidelines in the search for alternatives to General Relativity

Novel terminal region computation method for quasi-infinite horizon NMPC

An algorithm for invariant region characterization for a nonlinear system controlled by an LQR is proposed. The quasi-infinite horizon nonlinear model predictive controller formulation is extended for zone control with optimizing targets. The novel invariant region characterization proposed promotes hypervolume gains of up to two orders of magnitude for an unstable CSTR. Extension of the NMPC formulation to the case of zone control with optimizing targets improves the formulation’s practical deployment capability. A comparison between QIH-NMPC and NMPC with a terminal equality constraint is drawn, showing considerable closed-loop performance loss when employing a terminal equality constraint. The proposed invariant region shows feasibility set gains from the proposed invariant region characterization, when compared to a recent approach. Closed-loop simulations of both controllers from the enlarged feasibility set show how sensible the closed-loop performance is to one infeasible controller iteration.

Characterization of eventual conditional invariance for hybrid dynamical systems

Sufficient conditions to guarantee eventual conditional invariance (ECI) for hybrid dynamical systems are formulated. Roughly speaking, eventual conditional invariance encodes the property of solutions that, when they start from a given set of initial conditions, they reach a target set after a finite amount of time and remain in it for future time. The sufficient conditions for ECI presented in this paper are infinitesimal, in the sense that they need to be checked at points in the state space of the hybrid dynamical system under study, they involve its data and an appropriate choice of Lyapunov-like functions, and rely on solution properties of a continuous-time and a discrete-time scalar system. Rather than assuring finite-time attractivity plus forward invariance of the target set, as found in previous works, the proposed conditions allow the solutions to enter and leave the target set provided that they eventually settle in it after some time. Examples throughout the paper illustrate key concepts and results.

Characterization of Positive Invariance of Quadratic Convex Sets for Discrete-Time Systems Using Optimization Approaches

A positively invariant set is an important concept in dynamical systems. The study of positively invariant set conditions for discrete-time systems is one interesting topic in both theoretical studies and practical applications research. Different methods for characterizing the invariance of different types of sets have been established. For example, the ellipsoidal and the Lorenz cone, which are quadratic convex sets, have different properties from a polyhedral set. This paper presents an optimization method and a dual optimization method to characterize the positive invariance of the ellipsoidal and the Lorenz cone. The proposed methods are applicable to both linear and nonlinear discrete-time systems. Using nonlinear programming and an induced norm, the positive invariance condition problems are transformed into optimization problems, and the dual optimization method is also used to give equivalent dual forms. Fewer results on the positive invariance condition of Lorenz cones can be found than for the other type of set; this paper fulfills the results of this problem. In addition, the proposed methods in this paper provide more options for checking the positive invariance of quadratic convex sets from the perspective of optimization and dual optimization. The effectiveness of this method is demonstrated by numerical examples.

Semantic Document Layout Analysis of Handwritten Manuscripts

A document layout can be more informative than merely a document’s visual and structural appearance. Thus, document layout analysis (DLA) is considered a necessary prerequisite for advanced processing and detailed document image analysis to be further used in several applications and different objectives. This research extends the traditional approaches of DLA and introduces the concept of semantic document layout analysis (SDLA) by proposing a novel framework for semantic layout analysis and characterization of handwritten manuscripts. The proposed SDLA approach enables the derivation of implicit information and semantic characteristics, which can be effectively utilized in dozens of practical applications for various purposes, in a way bridging the semantic gap and providing more understandable high-level document image analysis and more invariant characterization via absolute and relative labeling. This approach is validated and evaluated on a large dataset of Arabic handwritten manuscripts comprising complex layouts. The experimental work shows promising results in terms of accurate and effective semantic characteristic-based clustering and retrieval of handwritten manuscripts. It also indicates the expected efficacy of using the capabilities of the proposed approach in automating and facilitating many functional, real-life tasks such as effort estimation and pricing of transcription or typing of such complex manuscripts.

Invariant Characterization of Liouville’s System with Two Degrees of Freedom

The local problem of finding separated generalized coordinates in a holonomic natural system with two degrees of freedom (if the solution exists) is reduced to the problem of integrability of the Pfaffian systems of differential equations.

Examining statistical isotropy of CMB low multipoles from Planck PR4 data

Low multipoles (l) in cosmic microwave background (CMB) temperature anisotropies have shown some ‘peculiarities’ when examined since the release of the full sky CMB maps, using a variety of tests. In this paper, we concern ourselves with the very first peculiarities seen in CMB data viz., a breakdown of statistical isotropy in the form of axiality and planarity of these low-l modes, and preferred alignments among them. We scrutinize the latest CMB data from ESA's Planck mission, PR4, to evaluate the current status of these deviations. We employ the Power tensor method which allows an invariant characterization of the distribution of power in a given multipole, and apply it to probe the first sixty multipoles i.e., l=2 to 61. We find that there are significant number of modes that are intrinsically anisotropic with a cumulative probability of 0.3%. However since the planarity study reveals that those modes that are unusually planar are subset of these anisotropic modes, we conclude that they may not be intrinsically planar. The quadrupole is still well aligned with the octopole. Besides, l=3, higher multipoles aligned with quadrupole are found to be insignificant. Interestingly, the collective alignment axis of the first sixty multipoles is found to be broadly closer to the axis of dipole, quadrupole, octopole and other modes aligned with l=2.

Invariant characterization of Szekeres models with positive cosmological constant

Invariant characterization of Szekeres models with positive cosmological constant

Fuzzy set-valued information systems and their homomorphisms based on data compression

A fuzzy set-valued information system (FSVIS) is a special information system (IS) where the value of an object under each attribute or each attribute value is a fuzzy set. Homomorphism is a powerful mathematical tool to deal with FSVISs, which can be used to study relationships among them. Based on data compression, we obtain some characterizations about FSVISs and their homomorphisms. First, some homomorphisms between FSVISs are introduced. After that, attribute reduction based on tolerance relation in a FSVIS is studied. Eventually, we get invariant characterizations of FSVISs based on some special homomorphisms under data compression.

Curvature invariants for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime

The curvature scalar invariants of the Riemann tensor are important in general relativity because they allow a manifestly coordinate invariant characterisation of certain geometrical properties of spacetimes such as, among others, curvature singularities, gravitomagnetism. We calculate explicit analytic expressions for the set of Zakhary–McIntosh curvature invariants for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime as well as for the Kerr–Newman–(anti-)de Sitter black hole. These black hole metrics belong to the most general type D solution of the Einstein–Maxwell equations with a cosmological constant. Explicit analytic expressions for the Euler–Poincare density invariant, which is relevant for the computation of the Euler–Poincare characteristic χ(M), and the Kretschmann scalar are also provided for both cases. We perform a detailed plotting of the curvature invariants that reveal a rich structure of the spacetime geometry surrounding the singularity of a rotating, electrically charged and accelerating black hole. These graphs also help us in an exact mathematical way to explore the interior of these black holes. Our explicit closed form expressions show that the above gravitational backgrounds possess a non-trivial Hirzebruch signature density. Possible physical applications of this property for the electromagnetic duality anomaly in curved spacetimes that can spoil helicity conservation are briefly discussed.

An Invariant Characterization of the Levi-Civita Spacetimes

In the years 1917–1919 Tullio Levi-Civita published a number of papers presenting new solutions to Einstein’s equations. This work, while partially translated, remains largely inaccessible to English speaking researchers. In this paper we review these solutions, and present them in a modern readable manner. We will also compute both Cartan–Karlhede and Carminati–Mclenaghan invariants such that these solutions are invariantly characterized by two distinct methods. These methods will allow for these solutions to be totally and invariantly characterized. Because of the variety of solutions considered here, this paper will also be a useful reference for those seeking to learn to apply the Cartan–Karlhede algorithm in practice.

Geometric surfaces: An invariant characterization of spherically symmetric black hole horizons and wormhole throats

We consider a spherically symmetric line element which admits either a black hole geometry or a wormhole geometry and show that in both cases the apparent horizon or the wormhole throat is partially characterized by the zero set of a single curvature invariant. The detection of the apparent horizon by this invariant is consistent with the geometric horizon detection conjectures and implies that it is a geometric horizon of the black hole, while the detection of the wormhole throat presents a conceptual problem for the conjectures. To distinguish between these surfaces, we determine a set of curvature invariants that fully characterize the apparent horizon and wormhole throat. Motivated by this result, we introduce the concept of a geometric surface as a generalization of a geometric horizon and extend the geometric horizon detection conjectures to geometric surfaces. As an application, we employ curvature invariants to characterize three important surfaces of the line element introduced by Simpson, Martin-Moruno, and Visser, which describes transitions between regular Vaidya black holes, traversable wormholes, and black bounces.

Gauge invariant target space entanglement in D-brane holography

It has been suggested in arXiv:2004.00613 that in Dp-brane holography, entanglement in the target space of the D-brane Yang-Mills theory provides a precise notion of bulk entanglement in the gravity dual. We expand on this discussion by providing a gauge invariant characterization of operator sub-algebras corresponding to such entanglement. This is achieved by finding a projection operator which imposes a constraint characterizing the target space region of interest. By considering probe branes in the Coloumb branch we provide motivation for why the operator sub-algebras we consider are appropriate for describing a class of measurements carried out with low-energy probes in the corresponding bulk region of interest. We derive expressions for the corresponding Renyi entropies in terms of path integrals which can be directly used in numerical calculations.

Geometric properties of a certain class of compact dynamical horizons in locally rotationally symmetric class II spacetimes

In this paper, we study the geometry of a certain class of compact dynamical horizons with a time-dependent induced metric in locally rotationally symmetric class II spacetimes. We first obtain a compactness condition for embedded [Formula: see text]-manifolds in these spacetimes, satisfying the weak energy condition, with non-negative isotropic pressure [Formula: see text]. General conditions for a [Formula: see text]-manifold to be a dynamical horizon are imposed, as well as certain genericity conditions, which in the case of locally rotationally symmetric class II spacetimes reduces to the statement that “the weak energy condition is strictly satisfied or otherwise violated”. The compactness condition is presented as a spatial first-order partial differential equation in the sheet expansion [Formula: see text], in the form [Formula: see text], where [Formula: see text] is the Gaussian curvature of [Formula: see text]-surfaces in the spacetime and [Formula: see text] is a real number parametrizing the differential equation, where [Formula: see text] can take on only two values, [Formula: see text] and [Formula: see text]. Using geometric arguments, it is shown that the case [Formula: see text] can be ruled out and the [Formula: see text] ([Formula: see text]-dimensional sphere) geometry of compact dynamical horizons for the case [Formula: see text] is established. Finally, an invariant characterization of this class of compact dynamical horizons is also presented.

Characterizations and uncertainty measurement of a fuzzy information system and related results

An information system (IS) is an important model in the field of artificial intelligence. A fuzzy information system (FIS) may be regarded as an IS under fuzzy environment. This paper obtains some results on a FIS. Operators on a FIS are first researched. Then, relationships between FISs are discussed from two aspects of dependence and separability, and dependence between FISs is characterized by the inclusion degree and two kinds of fuzzy distances between FISs are proposed. Next, algebraic characterizations of a FIS are obtained and invariant characterizations of a FIS under some homomorphisms are given. Finally, measuring uncertainty of a FIS is investigated, and experimental analyses illustrates that the proposed measures are suitable.

Event horizon detecting invariants

Some judiciously chosen local curvature scalars can be used to invariantly characterize event horizons of black holes in $D>3$ dimensions, but they fail for the three dimensional Ba\~nados-Teitelboim-Zanelli (BTZ) black hole since all curvature invariants are constant. Here we provide an invariant characterization of the event horizon of the BTZ black hole using the curvature invariants of codimension one hypersurfaces instead of the full spacetime. Our method is also applicable to black holes in generic dimensions but is most efficient in three, four, and five dimensions. We give four dimensional Kerr, five dimensional Myers-Perry and three dimensional warped-anti--de Sitter, and the three dimensional asymptotically flat black holes as examples.

An invariant characterization of the quasi-spherical Szekeres dust models

The quasi-spherical Szekeres dust solutions are a generalization of the spherically symmetric Lemaitre-Tolman-Bondi dust models where the spherical shells of constant mass are non-concentric. The quasi-spherical Szekeres dust solutions can be considered as cosmological models and are potentially models for the formation of primordial black holes in the early universe. Any collapsing quasi-spherical Szekeres dust solution where an apparent horizon covers all shell-crossings that will occur can be considered as a model for the formation of a black hole. In this paper we will show that the apparent horizon can be detected by a Cartan invariant. We will show that particular Cartan invariants characterize properties of these solutions which have a physical interpretation such as: the expansion or contraction of spacetime itself, the relative movement of matter shells, shell-crossings and the appearance of necks and bellies.

The equivalence problem for generic four-dimensional metrics with two commuting Killing vectors

We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the $2$-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means that the Killing leaves are not null, the metric is not orthogonally transitive (i.e., the distribution orthogonal to the Killing leaves is non-integrable), and two explicitly constructed scalar invariants $C_\rho$ and $\ell_{\mathcal C}$ are nonzero. All the invariants are designed to have tractable coordinate expressions. Assuming the existence of two functionally independent invariants, we solve the equivalence problem in two ways. As an example, we invariantly characterise the Van den Bergh metric. To understand the non-generic cases, we also find all $\Lambda$-vacuum metrics that are generic in the above sense, except that either $C_\rho$ or $\ell_{\mathcal C}$ is zero. In this way we extend the Kundu class to $\Lambda$-vacuum metrics. The results of the paper can be exploited for invariant characterisation of classes of metrics and for extension of the set of known solutions of the Einstein equations.

Asymptotics with a positive cosmological constant. IV. The no-incoming radiation condition

Consider compact objects --such as neutron star or black hole binaries-- in \emph{full, non-linear} general relativity. In the case with zero cosmological constant $\Lambda$, the gravitational radiation emitted by such systems is described by the well established, 50+ year old framework due to Bondi, Sachs, Penrose and others. However, so far we do not have a satisfactory extension of this framework to include a \emph{positive} cosmological constant --or, more generally, the dark energy responsible for the accelerated expansion of the universe. In particular, we do not yet have an adequate gauge invariant characterization of gravitational waves in this context. As the next step in extending the Bondi et al framework to the $\Lambda >0$ case, in this paper we address the following questions: How do we impose the `no incoming radiation' condition for such isolated systems in a gauge invariant manner? What is the relevant past boundary where these conditions should be imposed, i.e., what is the \emph{physically relevant} analog of past null infinity $\mathcal{I}^{-}_{0}$ used in the $\Lambda=0$ case? What is the symmetry group at this boundary? How is it related to the Bondi-Metzner-Sachs (BMS) group? What are the associated conserved charges? What happens in the $\Lambda \to 0$ limit? Do we systematically recover the Bondi-Sachs-Penrose structure at $\mathcal{I}^{-}_{0}$ of the $\Lambda=0$ theory, or do some differences persist even in the limit? We will find that while there are many close similarities, there are also some subtle but important differences from the asymptotically flat case. Interestingly, to analyze these issues one has to combine conceptual structures and mathematical techniques introduced by Bondi et al with those associated with \emph{quasi-local horizons}.