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The purpose of this paper is to review systematically the research of grey relation analysis (GRA) models.

Three different approaches, the springboard to build a GRA model, the angle of view in modelling, and the dimension of objects, are analysed, respectively.

The GRA models developed from the models based on relation coefficients of each point in the sequences in early days to the generalized GRA models based on integral or overall perspective. It evolved from the GRA models which measure similarity based on nearness, into the models which consider similarity and nearness, respectively. The objects of the research advanced from the analysis of relationship among curves to that among curved surfaces, and further to the analysis of relationship in three‐dimensional space and even the relationship among super surfaces in n‐dimensional space.

The further research on GRA models is proposed. One is about the property of GRA model. An in‐depth knowledge about the properties of GRA model will help people to understand its function, applicable area and requirements for modelling. The other one is about the extension of research object system. The object to be analysed should be extended from the common sequence of real numbers to grey numbers, vectors, matrices, and even multi‐dimensional matrices, etc.

The purpose is to present a new formal approach based on a partial integro‐differential equation, the space‐time state transition equation (STSTE), and on a set of general equations with which space‐time dynamical models of complex systems, such as social systems and ecosystems, can be built.

The STSTE provides the partial derivative of the density of a state‐variable with regard to time as a sum of time rates and space‐time rates. Time rates describe the dynamics of the system for each space‐point irrespectively of the other points, whilst space‐time rates describe this evolution as a consequence of the relation of each space‐point with a given set of other points of the space. This relation contains integrals over the accessibility domains (sets of space‐points with which each space‐point is related).

The STSTE is provided for any system of space‐coordinates and is compared with the reaction‐diffusion models (RD). The reason why it is more convenient to work with the STSTE than with the RD to model complex systems in the context of social systems and ecosystems is indicated.

An urban system (the city of Valencia, Spain) is presented as an application; an analytical solution strategy is stated under the simplest hypothesis for computing space‐time rates, and a computer program for the situation is developed to obtain numerical solutions.

A numerical comparison between the new STSTE model and the RD shows that, the STSTE model produces better results than the reaction diffusion model in validation.

The paper accepts the “functional paradigm” in the formative process of the image of an object. It holds that the used formulae must be mathematically demonstrated. As for many systems with memory, the frequently met formula is a defined integral. Its inferior limit is considered “the initial instant”. In this paper we show that: (1) The integral representation formula, that gives the image η (t) as a functional on the history ξ =ξ (τ),τ≤q t, of the evolutions both of the observed object and the observer, is only an approximate one. There exists the possibility to appreciate the order of size of the error. (2) The inferior limit of the integral may be mathematically determined and its choice requires some discussions. (3) A more precise formula of η (t) contains also a double integral upon a quadratic form of ξ(τ). The mathematical model is based on the differential calculus on the linear topological locally convex spaces.

Science of systems requires a specific and constructive mathematical model and language, which describe jointly such systemic categories as adaptation, self‐organization, complexity, evolution, and bring the applied tools for building a system model for each specific object of a diverse nature. This formalism should be connected directly with a world of information and computer applications of systemic model, developed for a particular object. The considered information systems theory (IST) is aimed at building a bridge between the mathematical systemic formalism and information technologies to develop a constructive systemic model of revealing information regularities and specific information code for each object.

To fulfill this goal and the considered systems' definition, the IST joins two main concepts: unified information description of interacted flows, initiated by the sources of different nature, with common information language and systems modeling methodology, applied to distinct interdisciplinary objects; general system's information formalism for building the model, which allows expressing mathematically the system's regularities and main systemic mechanisms.

The formalism of informational macrodynamics (IMD), based of the minimax variational principle, reveals the system model's main layers: microlevel stochastics, macrolevel dynamics, hierarchical dynamic network (IN) of information structures, its minimal logic, and optimal code of communication language, generated by the IN hierarchy, dynamics, and geometry. The system's complex dynamics originate information geometry and evolution with the functional information mechanisms of ordering, cooperation, mutation, stability, diversity, adaptation, self‐organization, and the double helix's genetic code.

The developed IMD's theoretical computer‐based methodology and the software has been applied to such areas as technology, communications, computer science, intelligent processes, biology, economy, management, and other nonphysical and physical subjects.

The IMD's macrodynamics of uncertainties connect randomness and regularities, stochastic and determinism, reversibility and irreversibility, symmetry and asymmetry, stability and instability, structurization and stochastization, order and disorder, as a result of micro‐macrolevel's interactions for an open system, when the external environment can change the model's structure.

This paper develops methods of Bayesian inference in a cointegrating panel data model. This model involves each cross-sectional unit having a vector error correction representation. It is flexible in the sense that different cross-sectional units can have different cointegration ranks and cointegration spaces. Furthermore, the parameters that characterize short-run dynamics and deterministic components are allowed to vary over cross-sectional units. In addition to a noninformative prior, we introduce an informative prior which allows for information about the likely location of the cointegration space and about the degree of similarity in coefficients in different cross-sectional units. A collapsed Gibbs sampling algorithm is developed which allows for efficient posterior inference. Our methods are illustrated using real and artificial data.

The n‐dimensional direct search algorithm, DIRECT, developed by Jones, Perttunen, and Stuckman has attracted recent attention from the multidisciplinary design optimization community. Since DIRECT only requires function values (or ranking) and balances global exploration with local refinement better than n‐dimensional bisection, it is well suited to the noisy function values typical of realistic simulations. While not efficient for high accuracy optimization, DIRECT is appropriate for the sort of global design space exploration done in large scale engineering design. Direct and pattern search schemes have the potential to exploit massive parallelism, but efficient use of massively parallel machines is non‐trivial to achieve. A fully‐distributed control version of DIRECT that is designed for massively parallel (distributed memory) architectures is presented. Parallel results are presented for a multidisciplinary design optimization problem – configuration design of a high speed civil transport.

To propose a novel method for defect reconstruction in electromagnetic non‐destructive testing (NDT).

The inversion method is based on an optimized database that contains the measured signals for some predefined defect prototypes. The database is supported by an anisotropic simplex mesh, which has been generated adaptively in the abstract n‐dimensional space, spanned by the model parameters of the defect type. The actual reconstruction reduces to a mesh search and interpolation. The described theory is demonstrated in the paper by a solved NDT test problem.

We have realized that in addition to sole defect reconstruction, the database provides meta‐information about the quality of the inversion, the suitability of the chosen defect model parameters, as well as the capabilities of the testing experiment.

Defect models having several parameters require a sophisticated mesh generation algorithm, which works in higher dimensions.

In the authors' opinion the mesh database approach offers a totally new point of view of a given inverse problem, and may help in the better understanding of its nature.

The author considers an invariant lightlike submanifold M, whose transversal bundle tr(TM) is flat, in an indefinite Sasakian manifold M¯(c) of constant φ¯-sectional curvature c. Under some geometric conditions, the author demonstrates that c=1, that is, M¯ is a space of constant curvature 1. Moreover, M and any leaf M′ of its screen distribution S(TM) are, also, spaces of constant curvature 1.

The author has employed the techniques developed by K. L. Duggal and A. Bejancu of reference number 7.

The author has discovered that any totally umbilic invariant ligtlike submanifold, whose transversal bundle is flat, in an indefinite Sasakian space form is, in fact, a space of constant curvature 1 (see Theorem 4.4).

To the best of the author’s findings, at the time of submission of this paper, the results reported are new and interesting as far as lightlike geometry is concerned.

This paper seeks to give an outline about the geometric concept of electronic circuits, where the jump behavior of nonlinear circuits is emphasized.

A sketch of circuit theory in a differential geometric setting is given.

It is shown that the structure of circuit theory can be given in a much better way than by means of a description of circuits using concrete coordinates. Furthermore, the formulation of a concrete jump condition is given.

In this paper, an outline is given about the state of the art of nonlinear circuits from a differential geometric point of view. Moreover, differential geometric methods were applied to two example circuits (flip flop and multivibrator) and numerical results were achieved.