WM groups and Ramsey theory

For amenable groups there are “correspondence principles” relating the behavior under group translation for sets of positive density to that of sets of positive invariant measure for ergodic actions of the group. When the group is also “minimally almost periodic” such actions are automatically weakly mixing; hence the name WM groups. We study here the combinatorial consequences of this state of affairs.     

Integrated decision support for aviation safety inspectors

This paper presents a systems viewpoint for developing an advanced decision support system for aircraft safety inspectors. Research results from a Federal Aviation Administration (FAA) sponsored project to use neural network and expert systems technology to analyze aircraft maintenance databases are summarized. One of the main objectives of this research is to define more refined “alert” indicators for national comparison purposes that can signal potential problem areas by aircraft type for safety inspector consideration.

Integration aspects are addressed on two levels: (1) integration of the various technical components of the decision support system, and (2) integration of the decision support system with individual behavior, management systems and organizational structure, as well as corporate culture across both formal and informal dimensions. The paper summarizes the creation of strategic “inspection profiles” for aging aircraft and reliability curve fitting for structural components both based upon using neural network technology. Also, the potential use of a model-based expert system to facilitate field inspection diagnostics is presented. Finally, a framework for developing an intelligent decision system to support aircraft safety inspections is proposed that links expert systems, neural networks, as well as a paradigm of the decision making process typically used in unstructured situations.     

Universal deformation formulas and breaking symmetry

We show that an algebra with a non-nilpotent Lie group of automorphisms or “symmetries” (e.g., smooth functions on a manifold with such a group of diffeomorphisms) may generally be deformed (in the function case, “quantized”) in such a way that only a proper subgroup of the original group acts. This symmetry breaking is a consequence of the existence of certain “universal deformation formulas” which are elements, independent of the original algebra, in the tensor algebra of the enveloping algebra of the Lie algebra of the group.     

The Marotto Theorem on planar monotone or competitive maps

In 1978, Marotto generalized Li–Yorke’s results on the criterion for chaos from one-dimensional discrete dynamical systems to n-dimensional discrete dynamical systems, showing that the existence of a non-degenerate snap-back repeller implies chaos in the sense of Li–Yorke. This theorem is very useful in predicting and analyzing discrete chaos in multi-dimensional dynamical systems. Yet, besides it is well known that there exists an error in the conditions of the original Marotto Theorem, and several authors had tried to correct it in different way, Chen, Hsu and Zhou pointed out that the verification of “non-degeneracy” of a snap-back repeller is the most difficult in general and expected, “almost beyond reasonable doubt,” that the existence of only degenerate snap-back repeller still implies chaotic, which was posed as a conjecture by them. In this paper, we shall give necessary and sufficient conditions of chaos in the sense of Li–Yorke for planar monotone or competitive discrete dynamical systems and solve Chen–Hsu–Zhou Conjecture for such kinds of systems.     

Multicomponent steady flows in porous media with phase transitions

Steady multiphase flow of a multicomponent mixture in a porous medium with phase transitions is considered. It is shown that, in a wide class of cases, the thermodynamic problem separates from the filtration problem and the latter is integrated in quadratures. The class of exact solutions which has been found is used to interpret indicator curves. Solutions are presented in an analytic form for systems of the “gas-condensate” and “oil-gas” type.     

A comparison of the monomial method and the S-system method for solving systems of algebraic equations

Two numerical methods for solving systems of equations have recently been proposed: a method based on monomial approximations (the “monomial method”) and a technique based on S-system methodology (the “S-system method”). The two methods have been shown to be fundamentally identical, that is, they are both equivalent to Newton's method operating on a transformed version of the system of equations. Yet, when applied specifically to algebraic systems of equations, they have significant computational differences that may impact the relative computational efficiency of the two methods. These computational differences are described. A combinatorial strategy for locating many, and sometimes all, solutions to a system of nonlinear equations has also been suggested previously. This paper further investigates the effectiveness of this strategy when applied to either of the two methods.     

Detecting simple dynamics in Cournot-like models

We introduced the so-called Cournot-like models, i.e. n-dimensional discrete dynamical systems which constitute the mathematical environment for modeling some economic and biological processes. The main aim of this work is to present a characterization of the dynamical simplicity for these types of systems through the property “to have zero topological entropy”. Cournot-like systems generalize the well-known economic situation of competition in a duopolistic market introduces by Cournot in 1838.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

Large localizations of finite groups

We construct examples of localizations in the category of groups which take the Mathieu group M₁₁ to groups of arbitrarily large cardinality which are “abelian up to finitely many generators.” The paper is part of a broader study on the group theoretic properties which are or are not preserved by localizations.     

Intrinsically parallel solution of systems of linear partial differential equations

We give here an intrinsically parallel algorithm, which solves systems of linear partial differential equations. The mathematical foundations of the algorithm rely upon a particular representation of polynomials on a structure called “hypercube”, introduced by Beauzamy-Frot-Millour, and use Bombieri's scalar product. This scalar product also permits a detailed study of the stability of the algorithm. Boundary conditions and compatibility conditions are handled by the algorithm in an intrinsically parallel manner. This algorithm has been implemented on a Connection Machine CM5, at the “Etablissement Technique Central de l'Armement” (Arcueil, France). We give here several numerical examples.     

Trajectories of lattice particles using the new approach to no integrability

We study two systems which lead to a lattice when an integration path is specified in “aesthetic field theory”. One of these cases involves nonsoliton type particles (magnitudes of maxima and minima oscillate in time). The other system is made up of soliton type particles. The two systems are intrinsically three-dimensional. We speak of the third dimension as “time”. In one of our solutions, the particles move on straight line trajectories, insofar as our numerical work indicates. In the other solution, the soliton type particles undergo what appears to be simple harmonic motion in both the x- and y-directions (loop motion). We then study these two systems using the new approach to integrability which involves a superposition principle and is characterized by a unique change function at each point. We still find multi maxima and minima. The systems are not as symmetric as the lattice. The soliton characteristic is preserved by the new method. We investigated the motion of lattice particles. We found evidence of maxima (minima) regions coalescing so that the location of the maxima (minima) became difficult to follow. The concept of location of particles may not even have a well-defined meaning here. We find examples of soliton particles appearing and disappearing. We conclude that the manner of integration in a no integrability theory can transform a system with well-defined trajectories into a system where particles can no longer be followed in time.     

Shadow block iteration for solving linear systems obtained from wavelet transforms

Matrices resulting from wavelet transforms have a special “shadow” block structure, that is, their small upper left blocks contain their lower frequency information. Numerical solutions of linear systems with such matrices require special care. We propose shadow block iterative methods for solving linear systems of this type. Convergence analysis for these algorithms are presented. We apply the algorithms to three applications: linear systems arising in the classical regularization with a single parameter for the signal de-blurring problem, multilevel regularization with multiple parameters for the same problem and the Galerkin method of solving differential equations. We also demonstrate the efficiency of these algorithms by numerical examples in these applications.     

Computation of the normal forms for general M-DOF systems using multiple time scales. Part I: autonomous systems

This paper is concerned with the symbolic computation of the normal forms of general multiple-degree-of-freedom oscillating systems. A perturbation technique based on the method of multiple time scales, without the application of center manifold theory, is generalized to develop efficient algorithms for systematically computing normal forms up to any high order. The equivalence between the perturbation technique and Poincaré normal form theory is proved, and general solution forms are established for solving ordered perturbation equations. A number of cases are considered, including the non-resonance, general resonance, resonant case containing 1:1 primary resonance, and combination of resonance with non-resonance. “Automatic” Maple programs have been developed which can be executed by a user without knowing computer algebra and Maple. Examples are presented to show the efficiency of the perturbation technique and the convenience of symbolic computation. This paper is focused on autonomous systems, and non-autonomous systems are considered in a companion paper.     

Stability of linear almost-Hamiltonian periodic systems

A linear Hamiltonian system with periodic coefficients is subject to a small “dissipative” perturbation that makes it asymptotically stable. The conditions under which the perturbation remains dissipative for all Hamiltonian systems sufficiently close to the original one are discussed.     

Systems of first-order chemical reactions

We consider the classical model in chemical kinetics of a system of n species in which each species is converted to every other species by a first-order reaction. Solutions to the initial-value problem are given in matrix form and the properties of the n × n matrix K representing the system are analysed. For arbitrary (i.e. non-negative) values of the first-order rate constants, zero is an eigenvalue, and the other eigenvalues are complex with negative real parts. Thus, in this case the system generally oscillates to equilibrium. However, if the principle of microscopic reversibility is applied, and if each species is converted directly to every other species, then the system cannot oscillate but must converge “exponentially” to equilibrium. We discuss when K is diagonalizable, and we calculate a bound for the eigenvalues of K. Special forms of K, corresponding to special systems of reactions, are also examined; these include reactions in the configuration of a “chain”, a “cycle”, a “node” and reactions comprising combinations of these. We find again that if the principle of microscopic reversibility is rigorously applied then oscillations cannot take place, but that if this principle is not applied then oscillations may take place. The system of rate equations considered can be used to model various chemical, physical and biological phenomena.     

None of the coronoid systems can be isometrically embedded into a hypercube

A graph that can be isometrically embedded into a hypercube is called a partial cube (or binary Hamming graph). Klavžar, Gutman and Mohar [S. Klavžar, I. Gutman, B. Mohar, Labeling of benzenoid systems which reflects the vertex-distance relations, J. Chem. Inf. Comput. Sci. 35 (1995) 590–593] showed that all benzenoid systems are partial cubes. In this article we show that none of the coronoid systems (benzenoid systems with “holes”) is a partial cube.     

Dynamics of an ecological model living on the edge of chaos

We present a new ecological model, which displays “edge of chaos” (EoC) in parameter space. This suggests that ecological systems are not chaotic, instead, their dynamics can be characterized as short-term recurrent chaos. The system’s dynamics is unpredictable and admits bursts of short-term predictability. We also provide results, which suggest that fully developed chaos will rarely be observed in natural systems.     

Occurrence and underlying mechanism of multi-stripe chaotic attractors

This paper is concerned with the generation of multi-stripe chaotic attractors. Simple periodic nonlinear functions are employed to transform the original chaotic attractors to a pattern with multiple “parallel” or “rectangular” stripes. The relationship between the system parameters related to some periodic functions and the shape of the generated attractor is analyzed. Theoretic analysis about the underlying mechanism of generating the parallel stripes in the attractors is given. A general creation mechanism of multi-stripe attractors of the Lorenz system and other well-known chaotic systems is derived from the proposed unified approach.     

An arithmetic of complete permutations with constraints, I: an exposition of the general theory

We develop an arithmetic of complete permutations of symmetric, integral bases; this arithmetic is comparable to that of perfect systems of difference sets with which there are several interrelations. Super-position of permutations provides the addition of this arithmetic. Addition if facilitated by complete permutations with a certain “splitting” property, allowing them to be pulled apart and reassembled. The split permutations also provide a singular direct product for complete permutations in conjunction with the multiplication (direct product) of the arithmetic which itself derives from that for perfect systems of difference sets.

We pay special attention to complete permutations satisfying constraints both fixed and variable; this is equivalent to embedding partial complete permutations in complete permutations. In the sequel, using this arithmetic, we investigate the spectra of certain constraints with respect to central integral bases which are of interest for the purpose of giving further constructions either of complete permutations with constraints or of irregular, extremel perfect systems of difference sets.     

Robust placement of sensors in dynamic water distribution systems

Designing a robust sensor network to detect accidental contaminants in water distribution systems is a challenge given the uncertain nature of the contamination events (what, how much, when, where and for how long) and the dynamic nature of water distribution systems (driven by the random consumption of consumers). We formulate a set of scenario-based minimax and minimax regret models in order to provide robust sensor-placement schemes that perform well under all realizable contamination scenarios, and thus protect water consumers. Single-and multi-objective versions of these models are then applied to a real water distribution system. A heuristic solution method is applied to solve the robust models. The concept of “sensitivity region” is used to visualize trade-offs between multiple objectives.