Construction of the value function in a pursuit-evasion game with three pursuers and one evader

A differential pursuit-evasion game is considered with three pursuers and one evader. It is assumed that all objects (players) have simple motions and that the game takes place in a plane. The control vectors satisfy geometrical constraints and the evader has a superiority in control resources. The game time is fixed. The value functional is the distance between the evader and the nearest pursuer at the end of the game. The problem of determining the value function of the game for any possible position is solved.

Three possible cases for the relative arrangement of the players at an arbitrary time are studied: “one-after-one”, “two-after-one”, “three-after-one-in-the-middle” and “three-after-one”. For each of the relative arrangements of the players a guaranteed result function is constructed. In the first three cases the function is expressed analytically. In the fourth case a piecewise-programmed construction is presented with one switchover, on the basis of which the value of the function is determined numerically. The guaranteed result function is shown to be identical with the game value function. When the initial pursuer positions are fixed in an arbitrary manner there are four game domains depending on their relative positions. The boundary between the “three-after-one-in-the-middle” domain and the “three-after-one” domain is found numerically, and the remaining boundaries are interior Nicomedean conchoids, lines and circles. Programs are written that construct singular manifolds and the value function level lines.     

Nonlinear observers for continuous fermentation processes

This paper deals with nonlinear observers for a microbial growth rate model in continuous fermentation processes. The nonlinear model is steered to an “extended” bilinear system by change of state variables and output injection. One particular case is presented for an analytical expression of the specific growth rate. Theoretical stability and convergence properties are also analyzed. The effectiveness of the proposed observers is illustrated on simulated data.     

Fuzzy logic in measurements

Our main interest in this paper is to translate from “natural language” into “system theoretical language”. This is of course important since a statement in system theory can be analyzed mathematically or computationally. We assume that, in order to obtain a good translation, “system theoretical language” should have great power of expression. Thus we first propose a new frame of system theory, which includes the concepts of “measurement” as well as “state equation”. And we show that a certain statement in usual conversation, i.e., fuzzy modus ponens with the word “very”, can be translated into a statement in the new frame of system theory. Though our result is merely one example of the translation from “natural language” into “system theoretical language”, we believe that our method is fairly general.     

Polygonal chains cannot lock in 4D

We prove that, in all dimensions d4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be achieved by algorithms that use polynomial time in the number of vertices, and result in a polynomial number of “moves”. These results contrast to those known for d=2, where trees can “lock”, and for d=3, where open and closed chains can lock.     

Comparing Fleming-Viot and Dawson-Watanabe processes

Fleming-Viot processes and Dawson-Watanabe processes are two classes of “superprocesses” that have received a great deal of attention in recent years. These processes have many properties in common. In this paper, we prove a result that helps to explain why this is so. It allows one to prove certain theorems for one class when they are true for the other. More specifically, we show that product moments of a Fleming-Viot process can be bounded above by the corresponding moments of the Dawson-Watanabe process with the same “underlying particle motion”, and vice versa except for a multiplicative constant. As an application, we establish existence and continuity properties of local time for certain Fleming-Viot processes.     

Mereological relations

A “mereological” relation is one of “part-to-whole” type. This paper proposes a model-theoretic approach to identifying and classifying such relations. Three initial applications of this approach are investigated. In each case, the class of identified relations is found to consist precisely of the partial orders on the underlying set. Further applications are proposed.     

A quantum mechanical approach to a fuzzy theory

Recently, we proposed a general measurement theory for classical and quantum systems (i.e., “objective fuzzy measurement theory”). In this paper, we propose “subjective fuzzy measurement theory”, which is characterized as the statistical method of the objective fuzzy measurement theory. Our proposal of course has a lot of advantages. For example, we can directly see “membership functions” (= “fuzzy sets”) in this theory. Therefore, we can propose the objective and the subjective methods of membership functions. As one of the consequences, we assert the objective (i.e., individualistic) aspect of Zadeh's theory. Also, as a quantum application, we clarify Heisenberg's uncertainty relation.     

Statistical learning control of uncertain systems: theory and algorithms

It has recently become clear that many control problems are too difficult to admit analytic solutions. New results have also emerged to show that the computational complexity of some “solved” control problems is prohibitive. Many of these control problems can be reduced to decidability problems or to optimization questions. Even though such questions may be too difficult to answer analytically, or may not be answered exactly given a reasonable amount of computational resources, researchers have shown that we can “approximately” answer these questions “most of the time”, and have “high confidence” in the correctness of the answers.     

A covariance inequality under a two-part dependence assumption

A covariance inequality is proved under a certain “two-part” dependence assumption. It generalizes and sharpens, with a simpler and more transparent proof, two earlier covariance inequalities used in central limit theory under certain “two-part” strong mixing assumptions.     

Multi-step quasi-Newton methods for optimization

Quasi-Newton methods update, at each iteration, the existing Hessian approximation (or its inverse) by means of data deriving from the step just completed. We show how “multi-step” methods (employing, in addition, data from previous iterations) may be constructed by means of interpolating polynomials, leading to a generalization of the “secant” (or “quasi-Newton”) equation. The issue of positive-definiteness in the Hessian approximation is addressed and shown to depend on a generalized version of the condition which is required to hold in the original “single-step” methods. The results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach (by comparison with “single-step” methods), particularly as the dimension of the problem increases.     

The Dempster–Shafer calculus for statisticians

The Dempster–Shafer (DS) theory of probabilistic reasoning is presented in terms of a semantics whereby every meaningful formal assertion is associated with a triple (p, q, r) where p is the probability “for” the assertion, q is the probability “against” the assertion, and r is the probability of “don’t know”. Arguments are presented for the necessity of “don’t know”. Elements of the calculus are sketched, including the extension of a DS model from a margin to a full state space, and DS combination of independent DS uncertainty assessments on the full space. The methodology is applied to inference and prediction from Poisson counts, including an introduction to the use of join-tree model structure to simplify and shorten computation. The relation of DS theory to statistical significance testing is elaborated, introducing along the way the new concept of “dull” null hypothesis.     

Classes of pretopological spaces closed under the formation of final structures

We investigate coreflective subconstructs of the construct Prtop of pretopological spaces and continuous maps and in particular the inclusion “order” between these subconstructs. We describe the smallest, second and third coreflective class and then all minimal elements that are strictly larger. Using these minimal elements we obtain a “partition” of the whole conglomerate of coreflective subconstructs of Prtop. The results dealing with classes in one member of this partition have an immediate interpretation in the framework of reflexive relations.     

Factors Influencing the Outcomes of Collaborative Mathematical Problem Solving: An Introduction

This study is an investigation of the factors that influence the effectiveness of collaboration on open-ended mathematical tasks. Students in Grades 3, 6, and 9 worked in groups of 3 on 2 chance and data tasks-1 related to fair dice and the other related to associations among variables presented on data cards. The groups' outcomes and the types of collaboration observed are investigated in relation to issues raised in the literature. Various phenomena are identified that influence cognitive “lifting,” “hovering,” and “falling,” that is, improvement, no change, and reduction in levels of functioning, respectively. These phenomena include cognitive factors, social or interpersonal factors, and external factors.     

Quasilinear conflict-controlled processes with additional restrictions

A class of conflict-controlled processes [1–3] with additional (“phase” type) restrictions on the state of the evader is considered. A similar unrestricted problem was considered in [4]. Unlike [5, 6] the boundary of the “phase” restrictions is not a “death line” for the evader. Sufficient conditions for the solvability of the pursuit and evasion problems are obtained, which complement a range of well-known results [5–10].^‡     

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.     

Interval orders based on weak orders

One definition of an interval order is as an order isomorphic to that of a family of nontrivial intervals of a linearly ordered set with [a,b] < [c,d] if b c. Fishburn's theorem states that an order is an interval order if and only if it has no four-element restriction isomorphic to the ordered set (shown in Fig. 1) “ ”. We show that an order is isomorphic to a family of nontrivial intervals of a weak order, ordered as above, if and only if it has no restriction to one of the four ordered sets (shown in Fig. 2) “ ”, a six-element crown or a six-element fence.     

One-dimensional, two-fluid modelling of turbulent premixed flames

The “two-fluid” mathematical model for turbulent combustion is applied to a one-dimensional, premixed, stabilized ducted flame. The flame is assumed to consist of two interspersed fluids (“reactants” and “products”), each characterized by its own properties and interacting through the exchange of mass, heat, and momentum. The distributions of pressure, densities, velocities, and volume fractions across the duct were successfully simulated. From a parametric study on the effects of the empirical constants involved in the interfluid relations, the significant dependence of the system on the parameters that characterize the mass transfer rate and the relative effect of mass transfer to momentum transfer was confirmed. The application of the model to transient states proved its ability to predict system oscillations.     

On behavior of two-dimensional non-plane parallel turbulent flows of inviscid gas

The time behavior of two-dimensional flows of inviscid gas in which the velocity component normal to the plane of independent variables and the vorticity components parallel to this plane are different from zero, is investigated. Equations of such flows form two different subsystems. The first subsystem describes a plane parallel (“primary”) flow without the third velocity component, and is independent of the second subsystem consisting of a single equation for the third velocity component and determining the “secondary” flow. The flows are analyzed with sufficient detail without using numerical integration which carries with it unavoidable errors, and without linearization, both of which are employed to a lesser or greater degree in the study of the evolution of vortical structures (see /1–6/).     

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.     

Stratification method in the theory of thin shells : PMM vol. 43, no. 6, 1979, pp. 1058–1064

It is shown that on the basis of certain simplifications induced in the physical and geometrical dependences, such a “stratification” of a shell can be achieved for which the fibers of each of two layers will be deformed just as thin rods whose axes agree with the lines of principal curvature of the shell middle surface. The approach to analyzing shells on the basis of the relationships to be obtained below is called the “stratification method”.