Narrow Results

The convection-controlled diffusion problem is hyperbolic in nature and its solutions tend to have numerical shocks. To solve the problem of time dependence, it is common to use finite-element method in the spatial region, while the algorithm proposed in the past with finite differential procedure is mostly limited to the fixed finite-element grid of the spatial region. We often need to use different finite-element spaces at different times, such as the spread of flame, oil and water frontier problems; so many mathematicians and engineers have set their sights on the use of dynamic finite-element space, but also put across a lot of dynamic finite-element methods in giving the general parabola problem of the variable-mesh finite-element method. The principal purpose of this paper is to adopt different spatial grids for different time layers, and project the approximate solution of the previous time layer to the present time layer to act as the initial value of the current layer to enable us deduce the stability at discrete times.

Discrete time SI and SIS epidemic models with vertical transmission are presented in this paper. With regard to the SI model with constant or variable population size, we introduce an epidemic threshold parameter, the basic reproductive number R₀, for predicting disease dynamics. R₀ > 1 implies that the disease tends to an endemic equilibrium, while R₀ < 1 implies disease extinction. On the other hand, for the SIS epidemic model with another form force of infection, the basic reproduction number R₀ determines the persistence or extinction of the disease. In the same time, we also explore the relationship between the demographic equation and the epidemic process. In particular, we show that the epidemic model can exhibit bistability (alternative stable equilibria) over a wide range of parameter values.

The purpose of this paper is to present some simple properties and applications of dynamic games with discrete time and a continuum of players. For such games relations between dynamic equilibria and families of static equilibria in the corresponding static games, as well as between dynamic and static best response sets are examined and an equivalence theorem is proven. The existence of a dynamic equilibrium is also proven. These results are counterintuitive since they differ from results that can be obtained in similar games with a finite number of players.

The theoretical results are illustrated with examples describing voting and exploitation of ecological systems.

In this paper we consider dynamic games with continuum of players which can constitute a framework to model large financial markets. They are called semi-decomposable games.

In semi-decomposable games the system changes in response to a (possibly distorted) aggregate of players' decisions and the payoff is a sum of discounted semi-instantaneous payoffs. The purpose of this paper is to present some simple properties and applications of these games. The main result is an equivalence between dynamic equilibria and families of static equilibria in the corresponding static perfect-foresight games, as well as between dynamic and static best response sets. The existence of a dynamic equilibrium is also proven. These results are counterintuitive since they differ from results that can be obtained in games with a finite number of players.

The theoretical results are illustrated with examples describing large financial markets: markets for futures and stock exchanges.

We consider the Leslie’s prey–predator model with discrete time. This model is given by a nonlinear evolution operator depending on five parameters. We show that this operator has two fixed points and define type of each fixed point depending on the parameters. Finding two invariant sets of the evolution operator, we study the dynamical systems generated by the operator on each invariant set. Depending on the parameters, we classify the dynamics between a predator and a prey of the Leslie’s model.

The f (R, T ) gravity theory was proposed as an extension of the f(R) theories, for which besides geometrical correction terms, proportional to the Ricci scalar R, one has also material correction terms, proportional to the trace of the energy-momentum tensor T. Those material extra terms prevent the energy-momentum tensor of the theory to be conserved. On the other hand, in the context of noncommutative quantum mechanics, the presence of compact dimensions whose coordinates do not commute with time imply that time evolution is discretized, a feature which induces violations of energy conservation. In the present work we propose a connection between these two effects, so that the energy nonconservation observed in the 4-dimensional f(R, T ) gravity can be understood as a macroscopic effect of nonconservative quantum transitions involving the compact extra dimension. It turns out that the energy flows between the ordinary (commutative) 4-dimensional spacetime and the compact extra dimension.

In the present paper, the classical semi-Markov model in discrete time is examined under the assumption of a fuzzy state space. The definition of a semi-Markov model with fuzzy states is provided. Basic equations for the interval transition probabilities are given for the homogeneous and non homogeneous case. The definitions and results for the fuzzy model are provided by means of the basic parameters of the classical semi-Markov model.

This chapter is an introduction to complexity theory (encompassing chaos — a subset of complexity), a nascent domain, although, it possesses a historical root. Some fundamental properties of chaos/complexity (including complexity mindset, nonlinearity, interconnectedness, interdependency, far-from-equilibrium, butterfly effect, determinism/in-determinism, unpredictability, bifurcation, deterministic chaotic dynamic, complex dynamic, complex adaptive dynamic, dissipation, basin of attraction, attractor, chaotic attractor, strange attractor, phase space, rugged landscape, red queen race, holism, self-organization, self-transcending constructions, scale invariance, historical dependency, constructionist hypothesis and emergence), and its development are briefly examined. In particular, the similarities (sensitive dependence on initial conditions, unpredictability) and differences between deterministic chaotic systems (DCS) and complex adaptive systems (CAS) are analyzed. The edge of emergence (2^nd critical value, a new concept) is also conceived to provide a more comprehensive explanation of the complex adaptive dynamic (CAD) and emergence. Subsequently, a simplified system spectrum is introduced to illustrate the attributes, and summarize the relationships of the various categories of common systems.

Next, the recognition that human organizations are nonlinear living systems (high finite dimensionality CAS) with adaptive and thinking agents is examined. This new comprehension indicates that a re-calibration in thinking is essential. In the human world, high levels of human intelligence/consciousness (the latent impetus that is fundamentally stability-centric) drives a redefined human adaptive and evolution dynamic encompassing better potentials of self-organization or self-transcending constructions, autocatalysis, circular causation, localized spaces/networks, hysteresis, futuristic, and emergent of new order (involving a multi-layer structure and dynamic) — vividly indicating that intelligence/consciousness-centric is extremely vital. Simultaneously, complexity associated properties/characteristics in human organizations must be better scrutinized and exploited — that is, establishing appropriate complexity-intelligence linkages is a significant necessity. In this respect, nurturing of the intelligence mindset and developing the associated paradigmatic shift is inevitable.

A distinct attempt (the basic strategic approach) of the new intelligence mindset is to organize around human intrinsic intelligence — intense intelligence-intelligence linkages that exploits human intelligence/consciousness sources individually and collectively by focusing on intelligence/consciousness-centricity, complexity-centricity, network-centricity, complexity-intelligence linkages, collective intelligence, org-consciousness, complex networks, spaces of complexity (better risk management <=> new opportunities <=> higher sustainability) and prepares for punctuation points (better crisis management <=> collectively more intelligent <=> higher resilience/sustainability) concurrently — illustrating the significance of self-organizing capability and emergence-intelligence capacity. The conceptual development introduced will serve as the basic foundation of the intelligent organization (IO) theory.