Invariant Characterization Research Articles

Joint Time–Frequency Scattering

In time series classification and regression, signals are typically mapped into some intermediate representation used for constructing models. Since the underlying task is often insensitive to time shifts, these representations are required to be time-shift invariant. We introduce the joint time-frequency scattering transform, a time-shift invariant representation which characterizes the multiscale energy distribution of a signal in time and frequency. It is computed through wavelet convolutions and modulus non-linearities and may therefore be implemented as a deep convolutional neural network whose filters are not learned but calculated from wavelets. We consider the progression from mel-spectrograms to time scattering and joint time-frequency scattering transforms, illustrating the relationship between increased discriminability and refinements of convolutional network architectures. The suitability of the joint time-frequency scattering transform for time-shift invariant characterization of time series is demonstrated through applications to chirp signals and audio synthesis experiments. The proposed transform also obtains state-of-the-art results on several audio classification tasks, outperforming time scattering transforms and achieving accuracies comparable to those of fully learned networks.

Set-invariance characterizations of discrete-time descriptor systems with application to active mode detection

In this paper, we study robust invariant set characterizations of discrete-time descriptor systems and propose an active mode detection mechanism for discrete-time descriptor systems considering multiple modes of operation. The considered class of descriptor systems assumes regularity, stability and is affected by unknown-but-bounded disturbances. As a first theoretical result, we establish a general framework for robust invariant sets for discrete-time descriptor systems in both causal and non-causal cases. Particular transformations are subsequently proposed for handling causal and non-causal descriptor systems and will be used to characterize the effects of disturbances. Based on these set-theoretic notions and a designed input signal for active set separations, we propose an active mode detection mechanism by exploiting the strong invariance properties.

Strong stability of linear parabolic time-optimal control problems

Sufficient conditions for strong stability of a class of linear time-optimal control problems with general convex terminal set are derived. Strong stability in turn guarantees qualified optimality conditions. The theory is based on a characterization of weak invariance of the target set under the controlled equation. An appropriate strengthening of the resulting Hamiltonian condition ensures strong stability and yieldsa prioribounds on the size of multipliers, independent of,e.g., the initial point or the running cost. In particular, the results are applied to the control of the heat equation into anL2-ball around a desired state.

Invariant characterizations of fuzzy information systems under some homomorphisms based on data compression and related results

A fuzzy information system is an important mathematical model in the field of artificial intelligence and the notion of homomorphisms between fuzzy information systems is a kind of tool to study data compression in fuzzy information systems. This paper investigates invariant characterizations of fuzzy information systems under some homomorphisms based on data compression. Upper-reducts, lower-reducts and reducts in a fuzzy information system are first investigated. Then, fuzzy sets in a fuzzy information system are divided into three categories according to their importance. Next, measures of uncertainty for fuzzy information systems are investigated and in order to show the performance of the proposed measures, a numerical experimental example on the Ecoli data set that comes from UCI Repository of machine learning databases is given. Finally, invariant and inverse invariant characterizations of a fuzzy information system under some homomorphisms are obtained.

A differential equation approach to implicit sweeping processes

In this paper, we study an implicit version of the sweeping process. Based on methods of convex analysis, we prove the equivalence of the implicit sweeping process with a differential equation, which enables us to show the existence and uniqueness of the solution to the implicit sweeping process in a very general framework. Moreover, this equivalence allows us to give a characterization of nonsmooth Lyapunov pairs and invariance for implicit sweeping processes. The results of the paper are illustrated with two applications to quasistatic evolution variational inequalities and electrical circuits.

Information structures in a covering information system

A covering information system as the generalization of an information system is an important model in the field of artificial intelligence. This paper explores information structures in a covering information system, and this kind of structure is viewed as a granular structure from the granular computing viewpoint. The concept of information structures in a covering information system is first described by means of set vectors. Then, relationships between information structures in a covering information system are studied from the two aspects of dependence and separation. Next, properties of information structures in a covering information system are given. Furthermore, invariant characterizations of covering information systems under homomorphisms are presented. Finally, as an application for information structures in a covering information system, the granularity measure of uncertainty for covering information systems is investigated. These results will be very helpful for establishing a framework for granular computing in information systems.

Chordal Komatu–Loewner equation for a family of continuously growing hulls

In this paper, we discuss the chordal Komatu–Loewner equation on standard slit domains in a manner applicable not just to a simple curve but also a family of continuously growing hulls. Especially a conformally invariant characterization of the Komatu–Loewner evolution is obtained. As an application, we prove a sort of conformal invariance, or locality, of the stochastic Komatu–Loewner evolution SKLE6,−bBMD in a fully general setting, which solves an open problem posed by Chen et al. (2017).

Invariant characterizations of fuzzy relation information systems under homomorphisms

A homomorphism is an important mathematical tool to study relationships between fuzzy relation information systems. This paper is devoted to investigating reduction in a fuzzy relation information system and its invariant characterizations under homomorphisms. Intersection-reduction, union-reduction, and reduction in a fuzzy relation information system are first proposed. Then, properties of intersection-reduction, union-reduction, and reduction are given. Next, fuzzy relations in a fuzzy relation information system are divided into necessary, relatively necessary, and unnecessary fuzzy relations according to the importance. Finally, some invariant and inverse invariant characterizations of fuzzy relation information systems under consistency and compatible homomorphisms are obtained, respectively. It is worth mentioning that by means of homomorphism, we can get the relatively smaller image system that has the same data structures (i.e., invariant characterizations) as a given original system.

Linearization of third‐order ordinary differential equations via point transformations

The linearization problem for scalar third‐order ordinary differential equations via point transformations was solved partially in the works of Al‐Dweik et al by the use of the Cartan equivalence method. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third‐order ordinary differential equation , which admits a four‐dimensional point symmetry Lie algebra. The invariant characterization is given in terms of function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained invariant is also presented. The approach provides auxiliary functions, which can be effectively utilized to determine the point transformation that does the reduction to the equivalent canonical form. Furthermore, illustrations to the main theorem and applications are given.

Invariant characterization of third-order ordinary differential equations [formula omitted] with five-dimensional point symmetry group

The Cartan equivalence method is applied to provide an invariant characterization of the third-order ordinary differential equation u″′=f(x,u,u′,u″) which admits a five-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained constant invariant is also presented. We also show how one obtains the point transformation that does the reduction to linear form. Moreover, some applications are provided.

Stochastic invariance of closed sets with non-Lipschitz coefficients

This paper provides a new characterization of the stochastic invariance of a closed subset of Rd with respect to a diffusion. We extend the well-known inward pointing Stratonovich drift condition to the case where the diffusion matrix can fail to be differentiable: we only assume that the covariance matrix is. In particular, our result can be applied to construct affine and polynomial diffusions on any arbitrary closed set.

Invariant characterization of scalar third‐order ODEs that admit the maximal point symmetry Lie algebra

The Cartan equivalence method is used to deduce an invariant characterization of the scalar third‐order ordinary differential equation , which admits the maximal 7‐dimensional point symmetry Lie algebra. The method provides auxiliary functions that can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation . Moreover, examples are given to illustrate the method.

Analysis of a Bianchi-like equation satisfied by the Mars-Simon tensor

The Mars-Simon tensor (MST), which, e.g., plays a crucial role to provide gauge invariant characterizations of the Kerr-NUT-(A)(dS) family, satisfies a Bianchi-like equation. In this paper, we analyze this equation in close analogy to the Bianchi equation, in particular it will be shown that the constraints are preserved supposing that a generalized Buchdahl condition holds. This permits the systematic construction of solutions to this equation in terms of a well-posed Cauchy problem. A particular emphasis lies on the asymptotic Cauchy problem, where data are prescribed on a space-like I (i.e., for ∧ > 0). In contrast to the Bianchi equation, the MST equation is of Fuchsian type at I, for which existence and uniqueness results are derived.

On Exact Solutions and Perturbative Schemes in Higher Spin Theory

We review various methods for finding exact solutions of higher spin theory in four dimensions, and survey the known exact solutions of (non)minimal Vasiliev’s equations. These include instanton-like and black hole-like solutions in (A)dS and Kleinian spacetimes. A perturbative construction of solutions with the symmetries of a domain wall is also described. Furthermore, we review two proposed perturbative schemes: one based on perturbative treatment of the twistor space field equations followed by inverting Fronsdal kinetic terms using standard Green’s functions; and an alternative scheme based on solving the twistor space field equations exactly followed by introducing the spacetime dependence using perturbatively defined gauge functions. Motivated by the need to provide a higher spin invariant characterization of the exact solutions, aspects of a proposal for a geometric description of Vasiliev’s equation involving an infinite dimensional generalization of anti de Sitter space are revisited and improved.

Invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression

Invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression

Curvature invariant characterization of event horizons of four-dimensional black holes conformal to stationary black holes

We introduce three approaches to generate curvature invariants that transform covariantly under a conformal transformation of a four dimensional spacetime. For any black hole conformally related to a stationary black hole, we show how a set of conformally covariant invariants can be combined to produce a conformally covariant invariant that detects the event horizon of the conformally related black hole. As an application we consider the rotating dynamical black holes conformally related to the Kerr-NUT-(Anti)-de Sitter spacetimes and construct an invariant that detects the conformal Killing horizon along with a second invariant that detects the conformal stationary limit surface. In addition, we present necessary conditions for a dynamical black hole to be conformally related to a stationary black hole and apply these conditions to the ingoing Kerr-Vaidya and Vaidya black hole solutions to determine if they are conformally related to stationary black holes for particular choices of the mass function. While two of the three approaches cannot be generalized to higher dimensions, we discuss the existence of a conformally covariant invariant that will detect the event horizon for any higher dimensional black hole conformally related to a stationary black hole which admits at least two conformally covariant invariants, including all vacuum spacetimes.

On a class of integrable systems of Monge-Ampère type

We investigate a class of multi-dimensional two-component systems of Monge-Ampère type that can be viewed as generalisations of heavenly type equations appearing in a self-dual Ricci-flat geometry. Based on the Jordan-Kronecker theory of the skew-symmetric matrix pencils, a classification of normal forms of such systems is obtained. All two-component systems of Monge-Ampère type turn out to be integrable and can be represented as the commutativity conditions of parameter-dependent vector fields. Geometrically, systems of Monge-Ampère type are associated with linear sections of the Grassmannians. This leads to an invariant differential-geometric characterisation of the Monge-Ampère property.

Algebraically special Einstein–Maxwell fields

The Geroch-Held-Penrose formalism is used to re-analyse algebraically special non-null Einstein-Maxwell fields, aligned as well as non-aligned, in the presence of a possible non-vanishing cosmological constant. A new invariant characterisation is given of the Garcia-Plebanski and Plebanski-Hacyan metrics within the family of aligned solutions and of the Griffiths metrics within the family of the non-aligned solutions. As a corollary also the double alignment of the Debever-McLenaghan 'class D' metrics with non-vanishing cosmological constant is shown to be equivalent with the shear-free and geodesic behaviour of their Debever-Penrose vectors.

Технология написания проектного кейса в жанре «кейкис»

Is there a necessary and sufficient set of qualitative and quantitative tools (phase and fractal), arranged in a specific sequence, and allowing yourself under time pressure and relatively fast to construct the project chosen for the case study for the problem situation? This research question in the article is given a positive answer. Ten years´ experience of writing individual and group research and educational research cases in the genre KEYKIS within teaching Graduate School of Corporate Management Russian Presidential Academy of National Economy and Public Administration, National Research University "Higher School of Economics" (HSE) and Business University MIRBIS several marketing and managerial disciplines, as well as an extensive consulting practice, allow us to make some generalizations. Show one possible set of qualitative and quantitative tools to build and visualize conceptual models and theories for private research specific cases. The presence of invariant characterization and measurement tools applicable for use on the adjacent flat hierarchies, and networks is showh. It has been shown that it is possible accurate and relatively quick to describe the complex poorly formalized interdisciplinary problematic situation with access to the field (tensor) of alternative solutions regarding adequate external challenges. Examples of practical application of presented equipment.

∗-reductions in a knowledge base

One of the advantages of rough set theory is the fact that an unknown target concept can be approximately characterized based on existing knowledge structures in a knowledge base. Knowledge reductions are one of the key problems in the study of rough set theory. In this paper, ∗-reductions and their features in a knowledge base are explored. Moreover, communication between knowledge bases is proposed based on the use of homomorphisms between relation information systems, and several invariant characterizations under homomorphisms are obtained. These results will be significant for establishing a framework for granular computing in knowledge bases.