A study on the structure of planning behaviour

Although there is an abundance of literature on planning, it is largely either about how we actually plan, about how we should plan, or about how we should organize our planning, and little effort seems to have been contributed to investigate how we actually organize our planning, which should become the basis for the above three kinds of research. In this paper, we consider first the relationship between planning and decision-making with a simple example (Section 2), and then the benefits and costs of planning more specifically (Section 3). After these preparatory considerations, we postulate a hypothesis which describes planning behaviour of individuals and organizations, with necessary definitions and assumptions (Section 4). Some propositions attained from the hypothesis are also included. For testing the validity of the hypothesis, an organism model of organization which deals with an organization as an open system in the environment is introduced (Section 5.1). We make additional assumptions to let the model plan (Section 5.2), and the behaviour of the model is generally supported by the practitioners and researcher interviewed (Section 5.3). Finally we discussed briefly the methodology employed and the possible applications of the hypothesis (Section 6).     

Necessary conditions for optimal controls for systems governed by parabolic partial delay-differential equations in divergence form with first boundary conditions

In this paper, we consider the question of necessary conditions for optimality for systems governed by second-order parabolic partial delay-differential equations with first boundary conditions. All the coefficients of the system are assumed bounded measurable and contain controls and delays in their arguments. The second-order parabolic partial delay-differential equation is in divergence form. In Theorem 4.1, we present results on the existence and uniqueness of weak solutions in the sense of Ladyzhenskaya-Solonnikov-Ural'ceva for this class of systems. An integral maximum principle and its point-wise version for the corresponding controlled system are established in Theorem 5.1 and Corollary 5.1, respectively.The authors wish to thank Dr. E. Noussair for his stimulating discussion and valuable comments in the preparation of this paper. Further, they also wish to acknowledge the referee of the paper for his valuable suggestions and comments. The discussion presented in Section 6 is in response to his suggestions.     

Variational problems with lags

In this paper, we study some questions concerning the minima of the functional $$J\left( y \right) = \int_{x_1 }^{x_2 } {f\left( {x,y\left( x \right),y\left( {x - r} \right),\dot y\left( x \right),\dot y\left( {x - r} \right)} \right)dx} $$ In Section 2, we obtain an analogue to the Jacobi condition to add to the list of previously obtained necessary conditions. A transversality condition is developed in Section 3. In Section 4, we obtain an existence theorem. The techniques used are modifications of those used in the classical problems. In Section 5, we show that the theory of fields for the classical problem fails to work for our problem.     

Linear complementarity problems with an invariant number of solutions

This paper studies the class INS of all realn × n matricesM for which the linear complementarity problem (q, M) has exactlyk solutions—k depending only onM—for all realn-vectorsq interior to the coneK(M) of vectors for which (q, M) has any solution at all. This generalizes the results in Cottle and Stone (1983) which deal with the subclassU in INS wherek equals one.After the first two sections of this paper, which introduce the problem and background material, we move on to examine necessary conditions for a matrixM to be in INS (Section 3) and sufficient conditions under whichM will be in INS (Section 4). Section 5 deals with the possible values whichk may have. Section 6 discusses related results concerning the geometry of linear complementarity problems. Finally, Section 7 deals with some known and new matrix classes which are in INS.     

Compatible connectedness in graphs and topological spaces

A topology on the vertex set of a graphG iscompatible with the graph if every induced subgraph ofG is connected if and only if its vertex set is topologically connected. In the case of locally finite graphs with a finite number of components, it was shown in [11] that a compatible topology exists if and only if the graph is a comparability graph and that all such topologies are Alexandroff. The main results of Section 1 extend these results to a much wider class of graphs. In Section 2, we obtain sufficient conditions on a graph under which all the compatible topologies are Alexandroff and in the case of bipartite graphs we show that this condition is also necessary.     

New light on Hensel''s lemma

The historical development of Hensel's lemma is briefly discussed (Section 1). Using Newton polygons, a simple proof of a general Hensel's lemma for separable polynomials over Henselian fields is given (Section 3). For polynomials over algebraically closed, valued fields, best possible results on continuity of roots (Section 4) and continuity of factors (Section 6) are demonstrated. Using this and a general Krasner's lemma (Section 7), we give a short proof of a general Hensel's lemma and show that it is, in a certain sense, best possible (Section 8). All valuations here are non-Archimedean and of arbitrary rank. The article is practically self-contained.     

Pure strategy equilibria in a class of systems defense games

This paper analyzes a class of systems defense games. Section 1 provides an overview. Section 2 reviews the assumptions used to specify a class of system defense games in Shubik and Weber (1978, 1981) and Shubik (1982). This is followed by a review of the additive games considered in Shubik and Weber (1981) — and the notion of comparable amounts of strategic resources used in that paper. Section 3 generalizes two of the features of additive system defense games with comparable resources. Section 4 states the final assumptions that are being made, and discusses the nature of the resulting class of systems defense games. The central result in the paper is in Section 5. The theorem specifically identifies an implicit objective function that is maximized by the equilibrium strategies. In Section 6, the theorem is used to also obtain an equilibrium existence result and identify conditions where there is a unique equilibrium.I gratefully acknowledge financial support provided by a Pew Grant to the University of Maryland for research on Economics and National Security. I also gratefully acknowledge helpful comments and suggestions provided by the participants in Maryland's Economics and National Security Luncheon Seminar and an anonymous referee.     

Functions of event variables of a random system with complete connections

In applications it often occurs that the experimenter is faced with functions of random processes. Suppose, for instance, that he only can draw partial or incomplete information about the underlying process or that he has to classify events for the sake of efficiency. We assume that the underlying process is a random system with complete connections (which contains the Markovian case as a special one) satisfying some basic properties, and that a mapping operates on the event space. With these two elements we construct in Section 2 a new random system with complete connections which inherits the properties of the old one (Theorem 2.2.3). In Section 3 we prove a weak convergence theorem (Theorem 3.4.4) in the theoretical framework of the so-called distance diminishing models, which gives a straightforward application in Section 4 to conditional probabilities related to partially observed events (Theorems 4.1.3). Finally we prove a Shannon-McMillan-type theorem (Theorem 4.2.3) finding application to classification procedures.     

Geometry in Grassmannians and a generalization of the dilogarithm

In this paper we introduce some polyhedra in Grassman manifolds which we call Grassmannian simplices. We study two aspects of these polyhedra: their combinatorial structure (Section 2) and their relation to harmonic differential forms on the Grassmannian (Section 3). Using this we obtain results about some new differential forms, one of which is the classical dilogarithm (Section 1). The results here unite two threads of mathematics that were much studied in the 19th century. The analytic one, concerning the dilogarithm, goes back to Leibnitz (1696) and Euler (1779) and the geometric one, concerning Grassmannian simplices, can be traced to Binet (1811). In Section 4, we give some of this history along with some recent related results and open problems. In Section 0, we give as an introduction an account in geometric terms of the simplest cases.     

The strong maximum principle revisited

In this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g.

(i)

two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4);

(ii)

the exterior Dirichlet boundary value problem (Section 5);

(iii)

the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7);

(iv)

Euler-Lagrange inequalities on a Riemannian manifold (Section 9);

(v)

comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10).

The case of p-regular elliptic inequalities is briefly considered in Section 11.     

Periodic solutions for coupled systems of differential-difference and difference equations

Summary The objective of this paper is to give necessary and sufficient conditions for the existence of periodic solutions of coupled systems of differential-difference and difference equations. By differentiating the difference equation, we obtain a system of neutral differential-difference equations and we get the original problem by putting a side condition on the neutral equation; that is, by restricting the initial data to lie on certain manifold in the space of all initial data. This allows us to treat the problem using the methods of neutral functional differential equations. In[8], Hale and Martinez-Amore exploited a certain change of variables to obtain some results on the stability of this systems. In Section 2, we summarize those ideas. The effect of the side condition is reflected in the variation of constants formula in Section 3. In this section, the variation of constants formula is decomposed via eigenspaces. In Section 4, we give a theorem on the Fredholm alternative for periodic solutions which is basic to the application of the usual theory to perturbed linear problems. I want to express my most deep gratitude to Professor J. K. Hale for his advice and suggestions which led to considerable improvements of this paper. Entrata in Redazione l'8 marzo 1978. Research was supported in the form of Grant from the Program of Cultural Exchange between the United States and Spain.     

Second-order conditions in extremal problems. The abnormal points

In this paper we study a minimization problem with constraints and obtain first- and second-order necessary conditions for a minimum. Those conditions - as opposed to the known ones - are also informative in the abnormal case. We have introduced the class of 2-normal constraints and shown that for them the ``gap" between the sufficient and the necessary conditions is as minimal as possible. It is proved that a 2-normal mapping is generic.

     

On Bayes estimates

In this paper the author tries to give general conditions for the existence of Bayes estimates and for the consistency of sequences of Bayes estimates.In Section 3 we prove existence theorems for Bayes estimates, which contain those of DeGroot and Rao [3], as a special case. The proof is based on a theorem of Landers [5].Section 4 gives a characterization of Bayes estimates with convex loss and linear decision space. This theorem is also a generalization of a similar theorem of DeGroot and Rao [3].In Section 5 we generalize the theory of minimum contrast estimates (the foundations of which were laid by Huber [4], cf. Pfanzagl [6]) in such a way that we can apply it to the theory of Bayes estimates.Section 6 tries to give a general theory of consistency for Bayes estimates using the martingale argument of Doob [1] and the theory of minimum contrast estimates. Confer in this connection the results of Schwartz [8].Section 7 contains some auxiliary results.     

Subdirect products of regular semigroups

A nonempty subset X contained in anH-class of a regular semigroup S is called agroup coset in S if XX′X=X and X′XX′=X′ where X′ is the set of inverses of elements of X contained in anH-class of S. Let μ denote the maximum idempotent separating congruence on S. We show in Section 1 of this paper that the set K(S) of group cosets in S contained in the μ-classes of S is a regular semigroup with a suitably defined product. In Section 2, we describe subdirect products of twoinductive groupoids in terms of certain maps called ‘subhomomorphisms’. A special class of subdirect products, called S^*-direct products, is described in Section 3. In the remaining two sections, we give some applications of the construction of S^*-direct products for describing coextensions of regular semigroups and for providing a covering theorem for pseudo-inverse semigroups.     

Nonstandard extensions of uniform algebraic systems

Important applications of nonstandard analysis to the theory of Banach spaces are based on the construction of the nonstandard hull of a normed linear space [1,2]. By making use of the iterated nonstandard enlargements [3], in the present article we propose a universal construction for arbitrary uniform algebraic systems which allows one to study the nonstandard hulls and completions of such systems.In Section 1 we give necessary information about nonstandard enlargements and prove a number of important results in the general theory of monads. In Section 2 we present basic facts on nonstandard topologies and uniform algebraic systems. In Section 3 we describe a general algebraic construction with the help of which we study the question of completing uniform algebraic systems. Conditions for existence of nonstandard hulls for such systems are obtained in Section 4.Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1094–1105, September–October, 1994.     

General models in min-max continous location: Theory and solution techniques

In this paper, a class of min-max continuous location problems is discussed. After giving a complete characterization of th stationary points, we propose a simple central and deep-cut ellipsoid algorithm to solve these problems for the quasiconvex case. Moreover, an elementary convergence proof of this algorithm and some computational results are presented.The work of the second author was supported by JNICT (Portugal), under Contract BD/631/90-RM, during his stay at Erasmus University in Rotterdam.The authors would like to thank the anonymous referees for simplifying the proofs in the first part of Section 2 and for their constructive remarks improving the presentation.     

Approximation Schemes for the Generalized Traveling Salesman Problem

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.     

Hereditarily strong lattices

Lower (or: dually) semimodular lattices behave in certain situations like consistent upper semimodular lattices. A common property of both classes of lattices is that they are hereditarily strong in the sense that every interval is a strong sublattice (cf. Section 2). We relate the properties “hereditarily strong” and “consistent” as well as their duals to lower semimodularity and to modularity (Section 3). Finally we characterize modularity for consistent lower semimodular lattices by means of forbidden sublattices (Section 4).     

Mean-field stochastic control with elephant memory in finite and infinite time horizon

ABSTRACT

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.

• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.

• For infinite horizon, we derive sufficient and necessary maximum principles.

  As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

     

Solving multi-objective production scheduling problems using metaheuristics

Most of research in production scheduling is concerned with the optimization of a single criterion. However the analysis of the performance of a schedule often involves more than one aspect and therefore requires a multi-objective treatment. In this paper we first present (Section 1) the general context of multi-objective production scheduling, analyze briefly the different possible approaches and define the aim of this study i.e. to design a general method able to approximate the set of all the efficient schedules for a large set of scheduling models. Then we introduce (Section 2) the models we want to treat––one machine, parallel machines and permutation flow shops––and the corresponding notations. The method used––called multi-objective simulated annealing––is described in Section 3. Section 4 is devoted to extensive numerical experiments and their analysis. Conclusions and further directions of research are discussed in the last section.