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In a recent paper entitled "Some sufficient conditions for occurring bifurcation" [1], H. S. Qin claims to have obtained sufficient conditions for bifurcation from multiple eigenvalues. The setting is that of an equation F(x,/~) = 0, (1) where F: R" x R--, R" is smooth, with F(0,/~) = 0, V# e R. It is assumed that for some Poe R we have rank DxF(O, ~) = n - rn, with 1 l, is misleading and gives a false impression that the results have to do with "higher multiplicities". What happens is that one looks for solutions in a subspace with codimension m - 1, while assuming that: (a) m - 1 of the n scalar equations in equation (1) are automatically satisfied for x in this subspace; (b) zero is a simple eigenvalue of DxF(O, ~o) when the nonrelevant dimensions are taken out of the picture. Condition (a) is especially severe when m > 1, except when it is a consequence of symmetry considerations (e.g. Ref. [3]). To be more precise let us formulate the following "generalized" version of the Crandall- Rabinowitz theorem. Theorem I Let X and Y be Banach spaces, X 0 a closed subspace of X and Y0 a closed subspace of Y. Let F: Y x R --* Y be a smooth mapping such that and

In this paper, a new Lagrangian function is reported which is particularly suited for large-scale nonconvex optimization problems with separable structure. Our modification convexifies the standard Lagrangian function without destroying its separable structure so that the primal-dual decomposition technique can be applied even to nonconvex optimization problems. Furthermore, the proposed Lagrangian results in two levels of iterative optimization as compared with the three levels needed for techniques recently proposed for nonconvex primal-dual decomposition.

After our paper [1], the central question about s -sum-sets is the following: does an s -sum-set that is not already a 3-sum-set exist? The main result of this paper gives a negative response to the question in the particular case where the s -sum-set is of type or 3.

A theoretical investigation of the propagation of pressure waves through uniformly converging and diverging thin-walled orthotropic elastic tubes filled with a Newtonian fluid is made in this paper, by considering the flow to be laminar and unsteady. Axisymmetric solutions of the differential equations that govern the motion of the fluid and the solid elastic wall are obtained. The outer surface of the wall is taken to be free of tractions. By using the solutions, a complicated form of transcendental equation is derived, that serves as a dispersion equation and puts forth a relation between the frequency and the wave number. This equation is solved by employing both analytical and numerical techniques. Numerical values of the pressure gradient, mean axial velocity, radial velocity, flow rate, amplitude ratio, phase velocity, and wave length have been computed for different angles of taper for a particular situation. It is felt that the results presented in the paper will find adequate applications in the study of blood flow through certain arterial segments, particularly in investigating the propagation of small amplitude harmonic waves, generated due to the flow of blood when the wave length is large compared to the radius of the arterial segment.

Nowadays, one-dimensional (1D) cellular automata (CA) based models and machines were demonstrated for various applications [1–7]. 2D CA based machines have not yet been used fruitfully mainly due to the lack of analytical insight into the behaviour of the underlying evolutionary process. The set of papers [8–11] dealt with the behaviour of the uniform 2D CAs. In the present paper, for the first time an attempt has been made to characterize a particular hybrid linear 2D CA transformation in GF (2) using matrix algebra. Also we deal with the synthesis problem of this particular hybrid transformation.

Reservoir operation is a special inventory problem under conditions of uncertain supply and controlled demand. In this paper, we will restrict our attention to reservoir operations. In many cases, the problem which the reservoir manager is concerned with is not to maximize the direct economic benefit, but to operate the reservoir system as normally as possible, i.e., to fulfill the requirement for water demand, recreation, fishing, generating electricity, and ecology, etc., and to avoid flooding, as much as possible. This paper presents a dynamic fuzzy criterion model (DFCM) for reservoir operations. In DFCM, a satisfactory degree function is adopted as a criterion function. The objective is to let the reservoir system be in the highest possible satisfactory state. This model is available to the reservoir system whose parameters are fuzzy or whose economical benefits are very difficult to measure. We obtain the existence, uniqueness, and stability theorems to the equation of DFCM, and prove that the optimal release policy for a reservoir is a bounded critical number policy. Finally, the application of DFCM in Qinhuangdao region water resource system is discussed.

The paper discusses a new, fully recursive approach to the adaptive modelling, forecasting and seasonal adjustment of nonstationary economic time-series. The procedure is based around a time variable parameter (TVP) version of the well known “component” or “structural” model. It employs a novel method of sequential spectral decomposition (SSD), based on recursive state-space smoothing, to decompose the series into a number of quasi-orthogonal components. This SSD procedure can be considered as a complete approach to the problem of model identification and estimation, or it can be used as a first step in maximum likelihood estimation. Finally, the paper illustrates the overall adaptive approach by considering a practical example of a U.K. unemployment series which exhibits marked nonstationarity caused by various economic factors.

Extending the results from the univariate case in a paper by Gal and Szabados, in this paper, we prove that the bivariate Shepard interpolation operators preserve the monotonicity and the convexity of bivariate functions in neighborhoods of some points.

In this paper, we deal with the problem of a fixed number of units of a certain perishable commodity over a continuous time horizon. Airline seats, hotel rooms, advertising space in newspapers, and some seasonal products that must be sold before the end of the season are examples of such commodities that cannot be carried over and are not storable for consumers. This paper considers such a problem of continuous time perishable inventory control by applying semi-Markov decision processes over a finite time horizon. It is shown that there is an optimal policy that is simple and stationary. Furthermore, some analytical properties of this optimal policy and its value function are explored under certain specific assumptions.

This paper introduces some results about stability for a class of linear point-delayed systems and how to compute these results. This work uses some theorems on stability for delayed systems due to Mori [1–3]. Our method consists basically in finding extensions of such results to the case of an approximate representation of the linear delayed system, i.e., under which circumstances stability for an approximate representation of the delayed system implies stability for the exact (real) delayed system. Sufficient conditions for asymptotic stability of linear delayed systems are greatly simplified and straightforwardly computed under Taylor series representation of the system state equations. Conditions under which asymptotic stability for the approximate system implies asymptotic stability for the real system are also outlined in the paper.

Software rejuvenation is a proactive fault management technique that has been extensively studied in the recent literature. In this paper, we focus on an example for a telecommunication billing application considered in [1] and develop the discrete-time stochastic models to estimate the optimal software rejuvenation schedules. More precisely, two software cost models with rejuvenation are formulated via the discrete semi-Markov processes, and the optimal software rejuvenation schedules which minimize the expected costs per unit time in the steady state are derived analytically. Further, we develop statistically nonparametric algorithms to estimate the optimal software rejuvenation schedules, provided that the complete sample data of failure times are given. Then, a new statistical device, called discrete total time on test statistics, is introduced. Finally, we examine asymptotic properties for the statistical estimation algorithms proposed in this paper through a simulation experiment.

In this paper, we consider solutions of Toeplitz systems Au = b where the Toeplitz matrices A are generated by nonnegative functions with zeros. Since the matrices A are ill-conditioned, the convergence factor of classical iterative methods, such as the Richardson method, will approach 1 as the size n of the matrices becomes large. In [1,2], convergence of the two-grid method with Richardson method as smoother was proved for band τ matrices and it was conjectured that this convergence result can be carried to Toeplitz systems. In this paper, we show that the two-grid method with Richardson smoother indeed converges for Toeplitz systems that are generated by functions with zeros, provided that the order of the zeros are less than or equal to 2. However, we illustrate by examples that the convergence results of the two-grid method cannot be readily extended to multigrid method for n that are not of the form 2 l − 1.

This paper discusses the lot size-reorder point inventory problem with fuzzy demands. Different from the existing studies, the shortages are backordered with shortage cost incurred. The α cut of the fuzzy demand is used to construct the fuzzy total inventory cost for each inventory policy ( Q , r ), where Q is the quantity to be ordered and r is the reorder point. Yager's ranking method for fuzzy numbers is utilized to find the best inventory policy in terms of the fuzzy total cost. Five pairs of simultaneous nonlinear equations for the optimal Q ∗ and r ∗ are derived for r in five different ranges of the fuzzy demand. When the demand is a trapezoidal fuzzy number, each pair of the simultaneous equations reduces to a set of closed-form equations. They are proved to be able to produce the optimal solution. Apparently, the methodology developed in this paper can be applied to other types of inventory problems to find the best inventory policy.

This paper is a geometric study of the local observer design for a general class of nonlinear systems with real parametric uncertainty. Explicitly, we study the observer design problem for a general class of nonlinear systems with real parametric uncertainty and with an input generator (exosystem). In this paper, we show that for the classical case, when the state equilibrium does not change with the parametric uncertainty, and when the plant output is purely a function of the state, there is no local asymptotic observer for the plant. Next, we show that in sharp contrast to this case, for the general case of problems where we allow the state equilibrium to change with the parametric uncertainty, there typically exist local exponential observers even when the plant output is purely a function of the state. We also present a characterization and construction procedure for local exponential observers for the general class of nonlinear systems with real parametric uncertainty under some stability assumptions. We also show that for the general class of nonlinear systems considered, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies, and is implied by, the existence of local exponential observers in the absence of inputs. Finally, we generalize our results to a general class of nonlinear systems with input generator, and with exogenous disturbance.

The main result of this paper is to show oscillations of nonlinear impulsive delay differential equations are equivalent to those of corresponding linear impulsive delay differential equations. The results of this paper generalize some well-known results in the literature.

Stochastic Automata Networks (SANs) are widely used in modeling practical systems such as queueing systems, communication systems, and manufacturing systems. For the performance analysis purposes, one needs to calculate the steady-state distributions of SANs. Usually, the steady-state distributions have no close form solutions and cannot be obtained efficiently by direct methods such as LU decomposition due to the huge size of the generator matrices. An efficient numerical method should make use of the tensor structure of SANs' generator matrices. The generalized Conjugate Gradient (CG) methods are possible choices though their convergence rates are slow in general. To speed up the convergence rate, preconditioned CG methods are considered in this paper. In particular, circulant based preconditioners for the SANs are constructed. The preconditioners presented in this paper are easy to construct and can be inverted efficiently. Numerical examples of practical SANs are also given to illustrate the fast convergence rate of the method.

This paper illustrates an example of a very fruitful effort of an industry research team in identifying and filling an analysis tool requirement in the area of composite bonded joints. The Composites Affordability Initiative, an Air Force sponsored joint aeronautics industry program, identified an analysis need for the rapid sizing of composite bonded joints. After an extensive review of the state of the art in modeling of bonded joints, a p-version finite-element-based commercial software was identified. This paper reviews the development history and describes new capabilities installed into this software through a joint effort of the CAI team and the software provider. The modifications implemented are discussed; and an example double-lap bonded joint analysis is presented. How the modeling philosophy of composite bonded joints differs from previous work is discussed, including the handling of singularities and error control. Actual failure criteria implemented, including stress/strain extraction logistics, are not reviewed, but examples of the general enhanced capabilities are discussed. An important lesson learned is the need for industry users to work closely with software providers to precisely tailor software to the requirements of the problem and the focus user group. This experience highlights the key link being forged between aerospace industry researchers and software providers to develop next generation analysis tools.

In this paper we deal with sensitivity analysis in multiobjective differential programs with equality constraints. We analyze the quantitative behavior of the optimal solutions according to changes of right-hand side values included in the original optimization problem. One of the difficulties lies in the fact that the efficient solution in multiobjective optimization in general becomes a set. If the preference of the decision maker is represented by a scalar utility which transforms optimal solutions in optima, we may apply existing methods of sensitivity analysis. However, when dealing with a subset of optimal points, the existence of a Fréchet differentiable selection of such a set-valued map is usually assumed. The aim of the paper is to investigate the derivative of certain set-valued maps of efficient points. We show that the sensitivity depends on a set-valued map associated to the T-Lagrange multipliers and a projection of its sensitivity.

To generate random numbers (RNs) of long period for large scale simulation studies, the usual multiplicative congruential RN generator can be extended to higher order. A multiplicative congruential RN generator of order two with prime modulus 231 − 1 attains a maximal period of (231 − 1)2 − 1 when the two multipliers (a1, a2) are chosen properly. By fixing a2 at −742938285, a multiplier recommended for first-order generator in a previous study, approximately 1.1 billion choices of a1 which are able to produce RNs of maximal period are investigated in this paper. Via the spectral test of dimensions up to six, 14 sets of multipliers (a1, a2) exhibit good lattice structure in a global sense with a spectral measure greater than 0.84. Ten of these multipliers also pass a battery of tests for detecting departures from local randomness and homogeneity. Furthermore, the execution time is promising on 32-bit machines. In sum, the second-order generators devised in this paper possess the properties of long period, randomness, homogeneity, repeatability, portability, and efficiency, required for practical use.

The paper gives a conceptual framework for robustness analysis of large-scale economic systems, and its realization through interactive computer-aided software. The mismatch between the economic system and the corresponding mathematical model is discussed. The computer-aided system combines algorithmic and expert system techniques. An important feature of the present system is the modularization of the software package which allows a distributed problem solving approach. A fourth-order macroeconomic model, with typical parameters, which demonstrates the margin of power of the governmental body can exercise on the various sectoral activities, is used to illustrate some of the concepts presented in this paper.

In this paper, we consider a class of nonlinear regulator problems in which the control appears linearly. Using an approach similar to that given for the classical linear quadratic regulator problem, it is shown in [1] that the optimal feedback control can be expressed as a function of the solution of an algebraic Riccati equation at each point in the state space. More precisely, it is shown that by solving a Riccati equation at a given point in the state space, the optimal feedback control at that particular point is readily obtained. In this paper, our first aim is to investigate stability of the resulting closed loop system. Second, a simple computational scheme for constructing a suboptimal control is suggested. We then consider the problem of stabilizing the system when it is subjected to bounded noise. For illustration, two examples are used to test the effectiveness of the proposed computational schemes.

This paper is a continuation of the preceding study [1] in which we described a method which automatically proves the existence of solutions for variational inequalities by computer. We newly formulate a verification method using a Newton-like method. This approach enables us to remove the restriction in the previous paper to the retraction property of the operator in a neighborhood of the solution. We show some numerical examples which confirm that the method is really applicable to problems which have no retraction property.

Maximum likelihood estimation of multivariate normal models and Bayesian posterior density functions can require calculation using numerical integration. Although the statistical theory is unchanged for nonanalytical integrals, the exorbitant cost of estimating such models frequently induces researchers to use a small number of integration points. If the number of points is insufficient to calculate essentially exact values of the function being maximized, the resulting estimators are asymptotically biased and inconsistent both for the values of the estimated parameters and for their variances. The forms of the biases are derived and described in this paper. The biases cannot be signed in general. The variance-covariance matrix is multiplied by an unestimable scalar, and the vector of true coefficients is added to an unestimable vector. When double integrals are involved, the number of points needed to calculate the integrals to desired accuracy is squared; with triple integrals, it is cubed, etc. The biases are additive, each integral contributing terms of similar form. An example of a multivariate normal model is presented. The paper shows that skimping on the high computational cost endangers the desirable statistical properties of MLE.

This paper deals with the construction of efficient algorithms for the solution of the finite dimensional constrained minimization problem arising from the finite element discretization of contact problems. Dualization techniques have been used to decrease the problem size from the large number of unknowns in the domain to the much smaller number of inequalities at the boundary. The disadvantage of this direct Schur-complement approach is the need of the inversion of the stiffness matrix. Domain decomposition techniques meet very similar requirements. A global boundary value problem is decoupled into local subproblems, and one interface problem at the (coupling) boundary. Two complementary approaches are the Dirichlet method and the Neumann method. The first one requires preconditioners for local Dirichlet problems and for the interface problem in H+ 1 2 , and extension operators from the boundary into the domain. The second one needs preconditioners for local Neumann problems, and for the interface problem in H− 1 2 . Efficient multilevel algorithms for all components are available in literature. In this paper, it is shown how to use exactly these components for the construction of solvers for contact problems. New results for the analysis of convergence are presented. At least at uniformly refined meshes, we can prove optimal time complexity. Numerical results show high efficiency also on adaptively refined 3D meshes.

In this paper, the variational iteration method (VIM) and the Adomian decomposition method (ADM) are implemented to give approximate solutions for fractional differential–algebraic equations (FDAEs). Both methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. This paper presents a numerical comparison between these two methods and the homotopy analysis method (HAM) for solving FDAEs. Numerical results reveal that the VIM and the ADM are quite accurate and applicable.

A new dyadic calculus is set up, principally based upon that introduced by Butzer and Wagner (1972/1975), and Zelin He (1983). The extended dyadic derivative of a function f, which is formulated for fractional orders, is roughly the Euler summation process applied to the Fourier-Walsh series of f after it has been equipped with a certain multiplicative factor. This extended calculus is not only applicable to piecewise constant functions (as is the classical dyadic derivative) but also to piecewise polynomials. This paper present the full general theory of this extended dyadic calculus: introduction and justification of the dyadic derivative, its fundamental properties and those of its eigenvalues; the corresponding anti-differentiation operator, a type of counterpart of the fundamental theorem of the Newton-Leibniz calculus in the frame of Walsh analysis. Applications and further theory are dealt with in the second paper in the series.

In 1901 Pearson formulated the general problem of how to fit a hyperplane in the most efficient way to a system of points in a data space. This problem is still not exactly solved in all its generality. As Kalman and Los have shown, all statistical attemps to solve the problem have failed, because each of them can provide only prejudicial and statistical, but not objective and mathematical solutions. However, exact mathematical solutions do exist for special cases. This paper's main principle of linear identification from inexact data provides the mathematical framework in which the problem and the deficiencies of the statistical solutions are conveniently discussed, in particular those of the least squares regression and statistical common factors schemes. It will be argued that the exact common factors, or Frisch scheme, offers most promise to direct us to complete and exact solutions, even though it imposes severe restrictions on the orders of the systems because of Wilson's inequality. Throughout this paper the problem and its various solution schemes are illustrated by an empirical example consisting of three data variables describing the profitability performance of some large U.S. bank holding companies. For this empirical example the Frisch scheme provides a unique solution, contrary to some earlier pessimistic conclusions.

In this paper, we propose and analyse a modified projection-type dynamical system associated with variational inequalities by using the technique of updating the solution. We prove that the globally asymptotic stability of this dynamical system requires only the pseudomonotonicity of the underlying operator, which is a weaker condition than monotonicity. The results obtained in this paper represent a significant improvement of the previously known results.

In this paper, we discuss the discontinuous quadratic finite element method for one-dimensional diffusion problems. We prove the stability for the polynomial basis functions of degree = 2, which is indicated by numerical experiments in the recent paper of Babuska, Baumann and Oden [1].

In this paper, we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite (a finite) family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. We obtain some weak convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, extend and unify some well-known weak convergence theorems in the literature.

In this paper, we study the static behavior of distributed memory architecture with general tree structures. After defining and discussing the notion of average diameter, we first show that the average time it takes to send messages between any two arbitrary processors in a binary tree structure with n processors is e(log n), through combinatorial analysis. We will also show that we can extend this result to general tree structures with n nodes, with a reasonable assumption, i.e., when m = o(n), where m is the maximum number of children any node could have. We believe that the results presented in this paper have captured some important inter-processor data transmission behavior of computers with distributed memory architectures, particularly those with (binary) tree structured inter-processor networks. As an application of combinatorial and a- symptotic analysis, this paper solves yet another average path length problem, thus is also theoreti- cally interesting. Finally, some of the techniques we have developed in this paper might also be useful in the study of similar problems. (~) 1998 Elsevier Science Ltd. All rights reserved.

A new mixed finite element method for the Dirichlet problem of fourth order elliptic equations with variable coefficients on convex polygonal domain has been developed in this paper, the biharmonic problem and bending problems of elastic plates being particular cases of the general problem considered in the paper. For bending problems, this method gives a simultaneous approximation to the displacement u and the ‘actual’ bending and twisting moment (ψij (i,j = 1, 2). Error estimate for the mixed finite element solution has been given.

In this paper, we study the appearance of limit cycles from the equator in a class of cubic polynomial vector fields with no singular points at infinity and the stability of the equator of the systems. We start by deducing the recursion formula for quantities at infinity in these systems, then specialize to a particular case of these cubic systems for which we study the bifurcation of limit cycles from the equator. We compute the quantities at infinity with computer algebraic system Mathematica 2.2 and reach with relative ease an expression of the first six quantities at infinity of the system, and give a cubic system, which allows the appearance of six limit cycles in the neighborhood of the equator. As far as we know, this is the first time that an example of cubic system with six limit cycles bifurcating from the equator is given. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of quantities at infinity is linear and then avoids complex integrating operations. Therefore, the calculation can be readily done with using computer symbol operation system such as Mathematica.

The fundamental problem in industrial robots control concerns algorithms generating reference trajectories. Papers [1–4] suggest generating algorithms of a reference trajectory, which are based on an arbitrary discretization of the manipulators internal coordinates. Each point of discretization in the external space approximating a reference trajectory corresponds with known discretized internal coordinates of the manipulator. In papers [5–7], iteration methods of determining the internal coordinates corresponding to external coordinates of the reference trajectory point have been suggested. In this method of internal coordinates, determining the point of the reference trajectory is being approached in successive steps of iterative computation. In paper [5], a modified iterative method of generating a reference trajectory straight segment has been presented. Analytic formulas which are the solution of an inverse problem of manipulator kinematics enable designing of trajectory generating algorithms which compute, in one step only, internal coordinates of points lying exactly on the reference trajectory, with the accuracy resulting from the computer register length. This paper presents equations of links and actuators kinematics of the IRb-6 manipulator in matrix form. Also, solutions of equations of link kinematics, as well as formulas joining link and actuator natural coordinates of the manipulator, have been presented.

The article of record as published may be located at http://dx.doi.org/10.1016/S0898-1221(99)00185-6 This paper analyzes the stability of the finite-element approximation to the linearized two-dimensional advection-diffusion equation. Bilinear basis functions on rectangular elements are considered. This is one of the two best schemes as was shown by Neta and Williams [1]. Time is discretized with the theta algorithms that yield the explicit (theta = 0), semi-implicit (theta = 1/2), and implicit (theta = 1) methods. This paper extends the results of Neta and Williams [1] for the advection equation. Giraldo and Neta [2] have numerically compared the Eulerian and semi-Lagrangian finite-element approximation for the advection-diffusion equation. This paper analyzes the finite element schemes used there. The stability analysis shows that the semi-Lagrangian method is unconditionally stable for all values of a while the Eulerian method is only unconditionally stable for 1/2 < theta < 1. This analysis also shows that the best methods are the semi-implicit ones (theta = 1/2). In essence this paper analytically compares a semi-implicit Eulerian method with a semi-implicit semi-Lagrangian method. It is concluded that (for small or no diffusion) the semi-implicit semi-Lagrangian method exhibits better amplitude, dispersion and group velocity errors than the semi-implicit Eulerian method thereby achieving better results. In the case the diffusion coefficient is large, the semi-Lagrangian loses its competitiveness. Published by Elsevier Science Ltd.

In this paper, we discuss the generalized ordering policy, in which when an ordered spare is delivered after a randomized lead time it is put into the inventory if an original unit is still operating and the original one is replaced/exchanged by the stocked spare when the original one fails/passes a prespecified time, whichever occurs first. We apply the expected total discounted cost as a criterion of optimality and obtain the optimal ordering policies minimizing this expected cost, taking into account several costs, an exponential-type discount rate and two types of randomized lead times. Finally, we present the relationships between the results of this paper and earlier contributions.

Singularly perturbed two-point boundary value problems (SPBVPs) for third-order ordinary differential equations (ODEs) with a small parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. Then, the computational method, presented in this paper, is applied to this system. In this method, we reduce the weakly coupled system into a decoupled system. Then, to solve this decoupled system numerically, we apply a ‘boundary value technique (BVT)’, in which the domain of definition of the differential equation is divided into two nonoverlapping subintervals called inner and outer regions. Then, we solve the decoupled system over these regions as two point boundary value problems. An exponentially fitted finite difference scheme is used in the inner region and a classical finite difference scheme, in the outer region. The boundary conditions at the transition point are obtained using the zero-order asymptotic expansion approximation of the solution of the problem. This computational method is distinguished by the facts that the decoupling reduces the computational time very much and it is well suited for parallel computing. This method can be extended to a system of two ordinary differential equations, of which, one is of first order and the other is of second order. Numerical examples are given to illustrate the method.

Among other representations, the relational databases are also widely used for storing spatial data. The model presented in this paper is a slightly modified version of the PLA database [1]. This spatial relational model serves to represent the topological properties of geographic data. In this paper, we first investigate the conditions for recognizing some types of the represented graph. We show how connectivity, 2-connectivity, Eulerian graphs, etc., can be characterized using just one relation of the database. Second, we point out redundancies in the representation and connections among the four relations of the database. Moreover, we design efficient (linear-time) algorithms for data retrieval/reconstruction of the stored spatial object, both in the planar and spherical cases. They also serve as constraints checking.

A generalization of Robbins-Monro stochastic approximation is presented in the paper. It is shown that, if disturbances satisfy a sort of generalized law of large numbers then an appropriate stochastic approximation procedure converges almost surely or only in probability, depending on what kind of law of large numbers (strong or weak) is satisfied by disturbances. In that sense theorems presented in the paper generalize Robbins-Monro stochastic approximation schemes, because the law of large numbers can be satisfied, as is well-known, by sequences of dependent random variables. On the other hand, as theorem of Gladishev (a generalized version of Robbins-Monro theorem) can be obtained from the results presented in the paper (see Theorem 10), one can consider this paper as the one providing new proofs for different versions of stochastic approximation. The proofs of the theorems of the paper are different than usual proofs of stochastic approximation procedure. In particular, they are not based on the Martingale convergence theorem. Roughly speaking the proofs exploit the analogy between the stochastic approximation procedures of Robbins-Monro versions and deterministic numerical iterative procedures seeking zeros of the system of nonliner equations. As the results of the paper were thought to be applied to estimation of parameters of discrete stochastic processes (so called identification) special notation has been introduced. This notation is believed to be useful for the above purpose.

This paper introduces a systematic approach for the derivation of models that describe the behavior of composites with a periodic structure. A technique for analyzing the accuracy of the models is addressed. Although the theory has been developed for the general, n-dimensional setting, the main ideas in this paper are presented for the 1-dimensional case only, for reasons of simplicity. The effectiveness of the approach is illustrated with numerical examples. Also, the approach has the potential of being used to adaptively select the best homogenization model.

In this paper, we shall discuss the relation between the expected cost model for a process and the probabilistic control model, whose mean is controlled by an X chart. T. Odanaka has exploited the problem of minimizing the membership function or the probability that the performance level of the machine will run out of range in a fuzzy environment. This paper considers the same basic problem of the expected cost model, modified to include the set up costs for controlling the action in any period. The expected cost comprises the fixed and variable cost of taking a control action, the cost of sampling, the cost of investigating the process when the control chart indicates that the process parameters exceed the specified bound, and the cost of producing defective units. Dynamic programming can be used to determine optimal control policies for models where control produces economically measurable benefits and/or costs. When the controlling action incurs a set up charge, the optimal policies are found to be characterized by a pair ( s n , S n , S ′ n , s ′ n ), where the action is made in a period n to state S n or S ′ n ( S n S ′ n ), if the state is found to be outside s n or s ′ n ( s n s ′ n ). This statement is subject to a set of basic assumptions such that proportional costs of changing the state variable are zero, the two fixed costs are equal, the loss function is symmetric quasi-convex and the problem's probability densities are quasi-concave.

In this paper, an analytical and numerical formulation for observed satellite radiances from space is provided, allowing for the radiation effect due to the adjacent rugged terrain. Up to now the atmospheric effect on the satellite images has been dealt with in the case of flat terrain. In this present paper, an allowance for the multiple reflection of light by the adjacent mountainous terrain has been taken into account approximately using the mean background albedo. Then, the validity of the numerical approximate solution is discussed for several cases of typical surface reflection examples, with respect to the atmospheric and topographic correction problem.

In this paper, we construct a new iterative algorithm for set-valued variational inclusions without the compactness condition and study the convergence of the perturbed Ishikawa iterative process for solving a class of the generalized single-valued variational inclusions in Banach spaces. The result obtained in this paper is a generalization and improvement of Noor's theorem [1].

The applicability of the group explicit (GE) methods to the numerical solution of hyperbolic partial differential equations of second-order is discussed in this paper following closely many of the ideas presented in an earlier related paper.

In this paper we should like to give a unified approach to the stochastic realization problem of finite dimensional discrete time stochastic systems. The generalized least squares innovation representation /GLS IR/ developed in the paper has diagonal error covariance matrix in the state space and orthonormed innovation process. Algorithm is presented for the identification of the parameters of the G L S IR.

Splitting methods are used to solve most of the linear systems, Ax=b, when the conventional method of Gauss is not efficient. These methods use the factorization of the square matrix A into two matrices M and N as A=M−N where M is nonsingular. Basic iterative methods such as Jacobi or Gauss–Seidel define the iterative scheme for matrices that have no zeros along its main diagonal.This paper is concerned with the development of an iterative method to approximate solutions when the coefficient matrix A has some zero diagonal entries. The algorithm developed in this paper involves the analysis of a discrete-time descriptor system given by the equation Me(k+1)=Ne(k), e(k) being the error vector.

In chemical physics, especially quantum mechanics, one encounters a myriad of phenomenological models based upon formal operator manipulations and calculations leading to spectral conclusions. Yet often the exact nature of such spectral results is unclear mathematically. On the other side, mathematical clarification can lead to greater physical insight. Two important illustrations of this symbiotic relationship are von Neumann's placing of the quantum mechanics of Schrodinger, Heisenberg, and others, into a Hilbert space framework, and Schwartz's rendering of the Dirac ket and delta notation into the theory of distributions, some thirty years later. With chemical physics now actively treating time-symmetry breakings into nonunitary evolutions, complex resonances, computations concerning the spectrum as caused by underlying chaotic dynamical systems, it seems useful to gather together here elements of operator spectral theory to provide a mathematical framework into which such spectral results can be placed or from which they may be viewed. A particularly useful, clarifying, but little-known device, is that of operator state diagram, which I will develop rather fully in this paper. All operator dualities and hence much of operator theory may be systematically portrayed by these diagrams. From these diagrams, I then discuss operator spectral states, scattering states available from the essential spectrum of an operator, spectral states of shift operators, and rigged Hilbert space spectral states. Only a few examples can be given here, for the rigorous working out of the state diagrams of a given new class of operators requires some care. Nonetheless, I hope that I will have provided some helpful clarifications of the instances given. Once the approach is seen, it is clear what needs to be done for other applications. Finally, I turn to what I shall call regular spectral states and probabilistic spectral states. These are new notions. Briefly, my view is that nature prefers regular spectral states, whereas we (humankind) prefer more specific spectral states. To reinforce this ansatz, I connect it to all other sections of this paper. A rule of probabilistically preferred spectral states is seen as an instance of the more general regularization principle which is an extensive irreversibility law. I include in each section a principle. This is meant to help the beginner come away with useful general notions, and it also permits one to express personal viewpoints.

This paper presents a possible generalization of geometric programming problems. Such a generalization was proposed by Peterson [6], based on Rockafellar's [8] conjugate function theory. Using their results, we define a slightly different, more symmetric dual pair of general unconstrained geometric programming problems. In the second chapter the conjugate function is defined and some of its properties are demonstrated. In the third chapter the general unconstrained geometric programming problem and its dual pair are introduced and some of its fundamental properties are proved. The primal optimality criteria is based on Peterson's papers [6,7] and the dual optimality criteria completes our examinations.

This paper is a sequel to Kurano [1,2], in which average cost Markov decision process (MDPs) with compact state and action spaces have been considered under the hypothesis of Doeblin and the existence of optimal stationary policies has been shown. The objective of this paper is to give the functional characterization. That is, we derive optimality inequalities for MDPs under the hypothesis of Doeblin and prove the validity of them under some reasonable conditions.