differentiable interpolants for high order runge kutta methods for a particular family of pairs of explicit runge kutta methods of orders p 1 and p sets of efficient continuously differentiable interpolants of several orders up to p are characters zed algorithmically in terms of several arbitrary parameters the approach can be applied in an obvious way to yield interpolants for other types of explicit runge kutta methods as well as families of other types of explicit and implicit methods derivative evaluations required for each pair of methods are reused and additional derivative evaluations are selected in an attempt to minimize the total number of stages required the analysis provides a lower bound on the number of stages required and indicates for example why twelve stages are required to provide interpolants for eight stage pairs of orders 5 and 6 in contrast to the eighteen stages used to obtain a known interpolant of order 7 for a pair of methods of orders 6 and 7 only sixteen stages are required by the proposed derivation details for interpola

flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

fuzzy differential games for nonlinear stochastic systems suboptimal approach a fuzzy differential game theory is proposed to solve the n person or n player nonlinear differential noncooperative game and cooperative game team problems which are not easily tackled by the conventional methods in the paper both noncooperative and cooperative quadratic differential games are considered first the nonlinear stochastic system is approximated by a fuzzy model based on the fuzzy model a fuzzy controller is proposed to deal with the noncooperative differential game in the sense of nash equilibrium strategies or with the cooperative game in the sense of pareto optimal strategies using a suboptimal approach the outcomes of the fuzzy differential games for both the noncooperative and the cooperative cases are parameterized in terms of an eigenvalue problem since the state variables are usually unavailable a suboptimal fuzzy observer is also proposed in this study to estimate the states for these differential game problems finally simulation examples are given to illustrate the design procedures and to indicate the performance of the proposed methods

uniformly superconvergent approximations for linear two point boundary value problems a local collocation scheme for solutions of first order systems of linear two point boundary value problems is developed that preserves uniformly the superconvergent accuracy at meshpoints achieved by collocation at gauss points over piecewise polynomials the method uses the superconvergent meshpoint values from the first collocation along with a second collocation that can be done independently on each piece a complete mathematical analysis is provided for the special cases when the points for the second collocation are equally spaced either open or closed it is shown that the choice of gauss points for this second collocation is always undesirable

numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

uncertain statistical inference models with imprecise observations the observations of some samples from the population of a probability density function with unknown parameters are usually imprecise due to various reasons by employing uncertain variables to model these imprecise observations this paper proposes an interdiscipline called the uncertain statistical inference which is composed of statistical inference and uncertainty theory it presents three types of statistic inference problems with imprecise observations that are the point estimation the hypothesis test and the interval estimation then it proposes a method of moments and a method of likelihood function maximum likelihood estimation for the first problem and a method of likelihood ratio function for the other two problems

linear quadratic differential games saddle point and riccati differential equation zhang siam j control optim 43 2005 pp 2157 2165 recently established the equivalence between the finiteness of the open loop value of a two player zero sum linear quadratic lq game and the finiteness of its open loop lower and upper values in this paper we complete and sharpen the results of zhang for the finiteness of the lower value of the game by providing a set of necessary and sufficient conditions that emphasizes the feasibility condition 0 0 is a solution of the open loop lower value of the game for the zero initial state then we show that under the assumption of an open loop saddle point in the time horizon 0 t for all initial states there is an open loop saddle point in the time horizon s t for all initial times s 0 le s t and all initial states at time s from this we get an optimality principle and adapt the invariant embedding approach to construct the decoupling symmetrical matrix function p s and show that it is an h 1 0 t solution of the matrix riccati differential equation thence an open loop saddle point in 0 t yields closed loop optimal strategies for both players furthermore a necessary and sufficient set of conditions for the existence of an open loop saddle point in 0 t for all initial states is the convexity concavity of the utility function and the existence of an h 1 0 t symmetrical solution to the matrix riccati differential equation as an illustration of the cases where the open loop lower upper value of the game is infty infty we work out two informative examples of solutions to the riccati differential equation with and without blow up time

a banzhaf function for a fuzzy game in this paper another important solution for fuzzy games called the banzhaf function is investigated first a sim plified expression for the banzhaf function for games with fuzzy coalition is proposed the relationship between this simplified ex pression and two kinds of particular banzhaf functions are dis cussed in detail furthermore in the framework of games with fuzzy coalitions we give a definition of the potential function of fuzzy games and prove that the marginal values of the potential function are the components of the banzhaf function further more a new axiomatic characterization of the banzhaf function for games with fuzzy coalitions is provided

an adaptive boundary value runge kutta solver for first order boundary value problems the use of finite difference methods which combine features of both runge kutta processes and gap schemes is proposed for developing adaptive codes for the solution of first order differential equations with two point boundary conditions the order conditions for the coefficients of these processes are given a set of conditions by which these order conditions can be reduced in number are developed an eighth order a stable method which has second fourth and sixth order a stable methods embedded in it is given as an example a variable order variable step finite difference solver using embedded methods of this kind is described

solving high order uncertain differential equations via runge kutta method high order uncertain differential equations are used to model differentiable uncertain systems with high order differentials and how to solve the high order uncertain differential equations is a core issue in practice this paper aims at proposing a numerical method to solve the high order uncertain differential equations based on the runge kutta recursion formula which is of high precision degree a procedure is designed and some numerical experiments are performed to illustrate the effectiveness and efficiency of the runge kutta method in addition this paper also presents a numerical method to calculate the expected value of a function of the solution

on computing the edge connectivity of an uncertain graph uncertain graphs are employed to describe graph models with imprecise expert data in uncertain graphs due to the existence of uncertain edges edge connectivity is essentially an uncertain variable different from that in a deterministic graph it is more suitable to investigate the possibility uncertain measure that an uncertain graph is k edge connected which is the main aim of this paper we first deduce a theoretical formula to calculate the uncertain measure and on this basis we then propose an algorithm which is derived from maximum flow algorithms to numerically calculate the uncertain measure the proposed algorithm is also proved to be a polynomial time algorithm and its effectiveness and efficiency are illustrated by numerical examples

optimal control of stochastic hybrid system with jumps a numerical approximation the generalized class of stochastic hybrid systems consists of models with regime changes including the occurrence of impulsive behavior in this paper the stochastic hybrid processes with jumps are approximated by locally consistent markov decision processes that preserve local mean and covariance we further apply a randomized switching policy for approximating the dynamics on the switching boundaries to investigate the validity of the approximation we study a stochastic optimal control problem on the basis of the discrete approximation we solve the optimal control problem and show that the solution obtained from the discretized problem converges to the solution of the original optimal control problem

uncertain alternating renewal process and its application uncertain process is a sequence of uncertain variables indexed by time and space first this paper presents a kind of uncertain process known as the uncertain alternating renewal process whose alternating interarrival times are uncertain variables then it proves an uncertain alternating renewal theorem on the limit value of average working rate finally an application of the alternating renewal theorem is discussed

shapley mappings and the cumulative value for n person games with fuzzy coalitions abstract in this paper we prove existence and uniqueness of the so called shapley mapping which is a solution concept for a class of n person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions the shapley mapping when it exists associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency the symmetry and the null player conditions it determines a cumulative value that is the sum of all coalitional allocations for whose computation we provide an explicit formula

some results on two person zero sum linear quadratic differential games considered in this paper are the two person zero sum linear quadratic differential games it is shown that the value of the game exists if and only if both the upper and lower values exist as a consequence we prove that another necessary and sufficient condition for the existence of the values of the game is the existence of an open loop open loop saddle point an example is also given in which the lower value exists but the upper value does not exist

uncertain calculus with yao process uncertain calculus is a branch of mathematics that deals with the differentiation and integration of functions of uncertain processes this paper investigates the yao process defined by the yao integral and extends the uncertain calculus on the yao process some important results are developed for finite variation and linearity of integral moreover this paper proposes a concept of the multifactor yao process and also gives a fundamental theorem for it

adaptive dynamic programming for discrete time zero sum games in this paper a novel adaptive dynamic programming adp algorithm called iterative zero sum adp algorithm is developed to solve infinite horizon discrete time two player zero sum games of nonlinear systems the present iterative zero sum adp algorithm permits arbitrary positive semidefinite functions to initialize the upper and lower iterations a novel convergence analysis is developed to guarantee the upper and lower iterative value functions to converge to the upper and lower optimums respectively when the saddle point equilibrium exists it is emphasized that both the upper and lower iterative value functions are proved to converge to the optimal solution of the zero sum game where the existence criteria of the saddle point equilibrium are not required if the saddle point equilibrium does not exist the upper and lower optimal performance index functions are obtained respectively where the upper and lower performance index functions are proved to be not equivalent finally simulation results and comparisons are shown to illustrate the performance of the present method

order barriers and characterizations for continuous mono implicit runge kutta schemes the mono implicit runge kutta mirk schemes a subset of the family of implicit runge kutta irk schemes were originally proposed for the numerical solution of initial value odes more than fifteen years ago dur ing the last decade a considerable amount of attention has been given to the use of these schemes in the numerical solution of boundary value ode prob lems where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes recent work in this area has seen the development of some software packages for bound ary value odes based on these schemes unfortunately these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval while the collocation schemes provide a natural continuous solution approximation the availability of a continuous solution is important not only to the user of the software but also within the code itself for example in estimation of errors defect control mesh selection and the provision of initial solution estimates for new meshes an approach for the construction of a continuous solution approximation based on the mirk schemes is suggested by recent work in the area of continuous extensions for explicit runge kutta schemes for initial value odes in this paper we describe our work in the investigation of continuous versions of the mirk schemes i we give some lower bounds relating the stage order to the minimal number of stages for general continuous irk schemes ii we establish lower bounds on the number of stages needed to derive continuous mirk schemes of orders 1 through 6 and iii we provide characterizations of these schemes having a minimal number of stages for each of these orders

stability in distribution for uncertain delay differential equation abstract as a type of differential equation uncertain delay differential equation is driven by liu process stability in measure stability in mean and stability in moment for uncertain delay differential equation have been proposed this paper mainly gives a concept of stability in distribution and proves a sufficient condition for uncertain delay differential equation being stable in distribution as a supplement moreover this paper further discusses their relationships among stability in distribution stability in measure stability in mean and stability in moment

uncertain wave equation with infinite half boundary wave equation is a type of second order and hyperbolic partial differential equation it is a commonly used tool to model many kinds of wave propagations such as sound wave electromagnetic wave water wave and string vibration propagations similarly uncertain wave equation is a type of uncertain partial equation driven by liu process which is widely used to model the wave propagation with uncertain noise such as vibrating string in uncertain environment the existing literature has studied uncertain wave equation with infinite boundary since infinite boundary is a much ideal condition this paper aims at studying the uncertain wave equation with infinite half boundary

some results about uncertain differential equations with time dependent delay abstract uncertain differential equations with time dependent delay are a type of differential equations driven by liu process this paper mainly proves that this type of uncertain differential equations have unique solutions in the infinite domain under some conditions in addition we also focus on the stability of uncertain differential equations with time dependent delay and some correspondence results are shown in the form of theorems

local piecewise polynomial projection methods for an o d e which give high order convergence at knots local projection methods which yield c sup m 1 piecewise polynomials of order m k as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functionals at which the residual error in each partition interval is required to vanish on we develop a condition on these k functionals which implies breakpoint superconvergence of derivatives of order less than m for the approximating piecewise polynomials the same order of superconvergence is associated with eigenvalue problems a discrete connection between two particular projectors yielding o vertical bardvertical bar sup 2k superconvergence namely a collocation at the k gauss legendre points in each partition interval and b essential least squares i e local moment methods is made by asking that this same order of superconvergence result when using collocation at k r points per interval and simultaneous local orthogonality of the residual to polynomials of order r the k r points then necessarily form a subset of the k gauss legendre points

runge kutta software with defect control four boundary value odes a popular approach to the numerical solution of boundary value ode problems involves the use of collocation methods such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval on the other hand several authors have suggested as an alternative certain subclasses of the implicit runge kutta formulas known as mono implicit runge kutta mirk formulas which can be implemented at a lower cost per step than the collocation methods these latter formulas do not have a natural implementation that provides a continuous approximation to the solution rather only a discrete approximation at certain points within the problem interval is obtained however recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit runge kutta formula these ideas have recently been extended to develop continuous extensions of the mirk formulas in this paper we describe our investigation of the use of continuous mirk formulas in the numerical solution of boundary value ode problems a primary thrust of this investigation is to consider defect control based on the continuous mirk formulas as an alternative to the standard use of global error control as the basis for termination and mesh redistribution criteria

shortest path problem with uncertain arc lengths uncertainty theory provides a new tool to deal with the shortest path problem with nondeterministic arc lengths with help from the operational law of uncertainty theory this paper gives the uncertainty distribution of the shortest path length also it investigates solutions to the a shortest path and the most shortest path in an uncertain network it points out that there exists an equivalence relation between the a shortest path in an uncertain network and the shortest path in a corresponding deterministic network which leads to an effective algorithm to find the a shortest path and the most shortest path roughly speaking this algorithm can be broken down into two parts constructing a deterministic network and then invoking the dijkstra algorithm

uncertain multi product newsboy problem with chance constraint a multi product newsboy problem with uncertain demand and uncertain storage space is investigated under a warehousing chance constraint it is assumed that indeterminacy may appear in demand and storage space due to rough estimation of their distributions by decision maker uncertainty theory provides a new tool to deal with the human uncertainty we develop an uncertain expected value programming model uevpm based on uncertainty theory uncertain variables functions from uncertainty space to the set of real numbers are used to describe subjective estimation the objective function is to maximize the expected profit of newsboy at a predetermined confidence level furthermore we convert the chance constrained uncertain programming model to its equivalent deterministic form so as to be solved by classic integer programming method finally three numerical examples are given to illustrate the potential usefulness of the study

order results for mono implicit runge kutta methods the mono implicit runge kutta methods are a subclass of the well known implicit runge kutta methods and have application in the efficient numerical solution of systems of initial and boundary value ordinary differential equations although the efficiency and stability properties of this class of methods have been studied in a number of papers the specific question of determining the maximum order of an s stage mono implicit runge kutta method has not been dealt with in addition to the complete characterization of some subclasses of these methods having a number of stages s leq 5 a main result of this paper is a proof that the order of an s stage mono implicit runge kutta method is at most s 1

reliability analysis of uncertain weighted k out of n systems the uncertain variable is used to model a quantity under human uncertainty and the weighted k out of n system is used to model a system of n components which functions if and only if the total weights of functioning components is greater than k considering the human uncertainty in operating the system this paper introduces the uncertain variable to the weighted k out of n system and proposes a concept of uncertain weighted k out of n system some formulas are derived to calculate the reliability index of such a system as a generalization this paper also studies an uncertain weighted k out of n system whose weights are estimated by experts and modeled by uncertain variables instead of crisp numbers in addition this paper analyzes the importance measure of the components in the uncertain weighted k out of n systems

fuzzy graphs in the evaluation and optimization of networks abstract referring to some predecessing models the paper introduces the concept of fuzzy vertex graphs a way is shown for modeling various quantitatively featurable functional capabilities of computer communication and similar networks it is shown that functional capability of distributed hierarchical multicomponent systems networks can be described by the directed rooted tree model according to the above fuzzy graph idea the problem of optimizing the overall functional capability with given component vertex set is introduced the fuzzy formulae based on the algebraic connectives which are to be optimized are constructed in the main part of the paper a series of results are presented concerning some special however important cases also the hypotheses founded on some computer experiments are formulated finally the application of these results and hypotheses are shown by the example of a distributed communication switching system family being in development in hungary

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

higher order approximations using interpolation applied to collocation solutions of two point boundary value problems when solving a system of first order differential equations involving boundary conditions using collocation at k gauss points that are the zeros of p k x the kth degree legendre polynomial to produce a continuous solution over the problem interval it is well known that the global error at the mesh points is o h 2 k and at other points the global error is o h k 1 in addition it is known that the error in the derivative of the solution at non mesh points is o h k this paper shows that the error in the solution at the zeros of p k x in each subinterval is o h k 2 and that the error in the derivative of the solution at the zeros of p k x is o h k 1 this higher order accuracy can be used in various interpolation schemes hermite birkhoff interpolation in the case that includes derivative data to produce global errors o h k 2 in the solution without any additional collocation and without any additional function evaluations only the evaluation of the collocation solution or its derivative at certain special points is necessary hence this increased accuracy is almost free compared to other methods which give global errors o h 2 k but at considerable extra expense some numerical results illustrating the higher accuracy at these special points are given at the end of the paper along with comparisons in the form of diagrams of results produced from the package colsys and the results produced from the interpolation schemes which use colsys as a starting point

system failure engineering and fuzzy methodology an introductory overview abstract system failure engineering may encompass reliability safety security maintainability risk and quality control in this paper we pay a unified view i e a failure oriented view to system failure engineering the notion of failure can be represented in terms of fuzzy sets and widely interpreted we then discuss various application aspects of fuzzy methodology in system failure engineering which include fuzzy methodology in fault diagnosis in probist systems in structural reliability in software reliability in human reliability in safety engineering in security engineering in risk engineering and in quality control as well as profust posbist and posfust reliability theories engineering case studies and mathematical problems raised by applications of fuzzy methodology in system failure engineering are also addressed in this paper

the maximum flow problem of uncertain network the maximum flow problem is one of the classic combinatorial optimization problems with many applications in electrical power systems communication networks computer networks and logistic networks the goal of the problem is to find the maximum amount of flow from the source to the sink in a network a network is called uncertain if the arc capacities of the network are uncertain variables the main purpose of this paper is to solve the maximum flow in an uncertain network by under the framework of uncertainty theory

on the numerical integration of nonlinear two point boundary value problems using iterated deferred corrections part 2 the development and analysis of highly stable deferred correction formulae iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two point boundary value problems in ordinary differential equations however there

an algorithmic approach to hermite birkhoff interpolation this paper deals with an algorithmic approach to the hermite birkhoff hb interpolation problem more precisely we will show that newton s classical formula for interpolation by algebraic polynomials naturally extends to hb interpolation hence almost all reasons which make newton s method superior to just solving the system of linear equations associated with the interpolation problem may be repeated let us emphasize just two newton s formula being a biorthogonal expansion has a well known permanence property when the system of interpolation conditions grows from newton s formula by an elementary argument due to cauchy an important representation of the interpolation error can be derived all of the above extends to hb interpolation with respect to canonical complete cebysev systems and naturally associated differential operators 7 a numerical example is given

numerical method for solving uncertain spring vibration equation abstract as a type of uncertain differential equations uncertain spring vibration equation is driven by liu process this paper proposes a concept of path and shows that the solution of an uncertain spring vibration equation can be expressed by a family of solutions of second order ordinary differential equations this paper also proves that the inverse uncertainty distribution of solution of uncertain spring vibration equation is just the path of uncertain spring vibration equation and a numerical algorithm is designed moreover a formula to calculate the expected value of solution of uncertain spring vibration equation is derived finally several numerical examples are provided to illustrate the efficiency of the numerical method

continuous numerical methods for odes with defect control abstract over the last decade several general purpose numerical methods for ordinary differential equations odes have been developed which generate a continuous piecewise polynomial approximation that is defined for all values of the independent variable in the range of interest for such methods it is possible to introduce measures of the quality of the approximate solution based on how well the piecewise polynomial satisfies the ode this leads naturally to the notion of defect control numerical methods that adopt error estimation and stepsize selection strategies in order to control the magnitude of the associated defect can be very effective and such methods are now being widely used in this paper we will review the advantages of this class of numerical methods and present examples of how they can be effectively applied we will focus on numerical methods for initial value problems ivps and boundary value problems bvps where most of the developments have been introduced but we will also discuss the implications and related developments for other classes of odes such as delay differential equations ddes and differential algebraic equations daes

renewal reward process with uncertain interarrival times and random rewards renewal reward process is used to measure the cumulative occasional rewards up to some given time so far two basic types of renewal reward processes with only random parameters or with only uncertain parameters have been proposed this paper aims at proposing a new type of renewal reward process which has uncertain interarrival times and random rewards in the framework of the chance theory the chance distribution of such an uncertain random renewal reward process is obtained and the reward rate is derived a renewal reward theorem is verified to show that the reward rate converges in distribution to an uncertain variable which is highly related to the interarrival times and the expected values of random rewards

indefinite stochastic linear quadratic control and generalized differential riccati equation a stochastic linear quadratic lq control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite indefinite stochastic lq theory has been extensively developed and has found interesting applications in finance however there remains an outstanding open problem which is to identify an appropriate riccati type equation whose solvability is it equivalent to the solvability of the indefinite stochastic lq problem this paper solves this open problem for lq control in a finite time horizon a new type of differential riccati equation called the generalized differential riccati equation is introduced which involves algebraic equality inequality constraints and a matrix pseudoinverse it is then shown that the solvability of the generalized riccati equation is not only sufficient but also necessary for the well posedness of the indefinite lq problem and the existence of optimal feedback open loop controls moreover all of the optimal controls can be identified via the solution to the riccati equation an example is presented to illustrate the theory developed

an error term and uniqueness for hermite birkhoff interpolation involving only function values and or first derivative values this paper discusses various aspects of hermite birkhoff interpolation that involve prescribed values of a function and or its first derivative an algorithm is given that finds the unique polynomial satisfying the given conditions if it exists a mean value type error term is developed which illustrates the ill conditioning present when trying to find a solution to a problem that is close to a problem that does not have a unique solution the interpolants we consider and the associated error term may be useful in the development of continuous approximations for ordinary differential equations that allow asymptotically correct defect control expressions in the algorithm are also useful in determining whether certain specific types of problems have unique solutions this is useful for example in strategies involving approximations to solutions of boundary value problems by collocation

fuzzy universal generating functions for multi state system reliability assessment the paper considers a multi state system mss where performance rates or and corresponding state probabilities are presented as fuzzy values in many real world msss it is difficult to estimate precise values of performance rates and their probabilities because of two main reasons the first reason is inaccuracy and insufficiency of data the second one is based on the fact that msss are often used as the approximation for continuous state systems in order to simplify computational burden to overcome these deficiencies performance rates levels or and corresponding probabilities are considered as fuzzy values in this paper the fuzzy universal generating function fugf is developed to extend the universal generating function with crisp sets the composition operators are extended for fugfs and introduced in more general forms based on this a special technique for reliability assessment of such msss is developed illustrative examples are presented to demonstrate the technique

solving uncertain heat equation via numerical method abstract uncertain heat equation is a type of uncertain partial differential equations driven by liu processes this paper proposes a concept of path for uncertain heat equation and shows that the solution of an uncertain heat equation can be represented by a family of solutions of ordinary heat equations and a formula is derived to calculate expected value of solution of uncertain heat equation moreover a numerical method is designed to solve uncertain heat equation several examples are given to illustrate the efficiency of the numerical method

fuzzy nash equilibriums in crisp and fuzzy games in this paper we introduce fuzzy nash equilibrium to determine a graded representation of nash equilibriums in crisp and fuzzy games this interpretation shows the distribution of equilibriums in the matrix form of a game and handles uncertainties in payoffs in addition a new method to rank fuzzy values with the user s viewpoint is investigated by this means the definition of satisfaction function which provides the result of comparison in the form of real value is developed when users have preferences regarding the payoffs

l2 cohomology of arithmetic varieties abstract the l2 cohomology of arithmetic quotients of bounded symmetric domains is studied we establish the conjecture of zucker equating the l2 cohomology of these spaces to the intersection cohomology of their baily borel compactifications

on distribution function of the diameter in uncertain graph in uncertain graphs the existence of some edges is not predetermined the diameter of an uncertain graph is essentially an uncertain variable which indicates the suitability for investigation of its distribution function the main focus of this paper is to propose an algorithm to determine the distribution function of the diameter of an uncertain graph we first discuss the characteristics of the uncertain diameter and the distribution function is derived an efficient algorithm is designed based on floyd s algorithm further some numerical examples are illustrated to show the efficiency and application of the algorithm

cooperative fuzzy games we derive versions of known results dealing with core equilibria and shapley values of cooperative games in the case of cooperative fuzzy games i e games defined on fuzzy subsets of the set of n players a fuzzy coalition is an n vector i associating with each player i his rate of participation i 0 1 in the fuzzy coalition and the real number v is the coalition worth assumed to be positively homogeneous if also v is superadditive or equivalently concave the fuzzy game with side payments has a nonempty core associated with the coalition worth function for any ordinary game with side payments is a fuzzy extension thereof viz the fuzzy game whose coalition worth function is the least positively homogeneous superadditive function majorizing the coalition worth function of the original game the fuzzy extension always has a nonempty core moreover if the original game has a nonempty core it coincides with that of its fuzzy extension analogous results are established for games without side payments an axiomatization of values of fuzzy games with side payments is also given the results are applied to show that the set of walras equilibria coincides with the fuzzy core of an economy

a stable high order interpolation scheme for superconvergent data a local collocation scheme is developed that yields stable high order accurate interpolants of discrete data arising from the numerical solution of a differential equation it should prove to be especially attractive for applications where data are superconvergent e g spline collocation at gauss points for simplicity the formulas are initially developed for a scalar equation but generalizations are later given for systems numerical examples are shown that illustrate the stability even for the case of highly nonuniform meshes which have proven difficult in prior studies

linear quadratic stochastic differential games open loop and closed loop saddle points in this paper we consider a linear quadratic stochastic two person zero sum differential game the controls for both players are allowed to appear in both drift and diffusion of the state equation the weighting matrices in the performance functional are not assumed to be definite non singular the existence of an open loop saddle point is characterized by the existence of an adapted solution to a linear forward backward stochastic differential equation with constraints together with a convexity concavity condition and the existence of a closed loop saddle point is characterized by the existence of a regular solution to a riccati differential equation it turns out that there is a significant difference between open loop and closed loop saddle points also it is found that there is an essential feature that prevents a linear quadratic optimal control problem from being a special case of linear quadratic two person zero sum differential games

uncertain random alternating renewal process with application to interval availability as a mixture of a random variable and an uncertain variable an uncertain random variable is a tool to deal with the indeterminacy quantities involving randomness and human uncertainty this paper aims at proposing a concept of an uncertain random alternating renewal process to model a repairable system with random on times and uncertain off times an alternating renewal theorem is proved which gives the limit chance distribution of the interval availability of the uncertain random alternating renewal system

single period inventory problem under uncertain environment this paper investigates the single period inventory newsboy problem in a general uncertain environment here it is assumed that the uncertainty appears in the market demand of the product uncertain demand is used to describe a subjective estimate not estimated by random fuzzy demand linguistically expressed by demand is about x using identification functions of the uncertain variable we derive the optimum order quantity and finally the expression of total expected profit for a fixed time period the computational aspects of the model in uncertain environment and their interpretations are illustrated by some examples

high order interpolants for solutionsof two point boundary value problems using mirk methods in this paper high order interpolants are presented for constructing continuous solutionsto a system of two point boundary value differential equations between widely spaced but accurate dependent variable values these interpolants are local and symmetric requiring data only within a single mesh interval and they require a small number of right hand side evaluations of the defining ode system to achieve the required order of accuracy internal derivative information in the mono implicit runge kutta formulae is exploited to reduce the number of additional right hand side evaluations necessary to define the interpolant to the required order of accuracy when the underlying ode system is second order very economical and accurate interpolants are found all of the interpolants are suitable for grid refinement algorithms in automatic adaptive two point boundary value packages such as twpbvp

least squares estimation in uncertain differential equations uncertain differential equations are a type of differential equations driven by liu processes how to estimate the parameters in an uncertain differential equation based on the observed data is a crucial problem in the real applications of these equations by means of the least squares estimation this article proposes a principle of minimum noise as an approach to the problem following this principle the estimates of the parameters in some special types of uncertain differential equations are derived which are represented as functions of the observed data in addition some numerical experiments are performed to illustrate the principle

superconvergent interpolants for the collocation solution of boundary value ordinary differential equations a long standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation obtained at the mesh points which subdivide the problem interval it is well known that the use of collocation at gauss points leads to solution approximations at the mesh points for which the global error is o h2k where k is the number of collocation points used per subinterval and h is the subinterval size this discrete solution is said to be superconvergent the collocation solution also yields a c0 continuous solution approximation that has a global error of o hk 1 in this paper we show how to efficiently augment the superconvergent discrete collocation solution to obtain c1 continuous superconvergent interpolants whose global errors are o h2k the key ideas are to use the theoretical framework of continuous runge kutta schemes and to augment the collocation solution with inexpensive monoimplicit runge kutta stages specific schemes are derived for k 1 2 3 and 4 numerical results are provided to support the theoretical analysis

the relative efficiency of alternative defect control schemes for high order continuous runge kutta formulas the development of continuous runge kutta formulas for nonstiff problems has been the subject of several investigations in recent years in enright siam j numer anal 26 1989 pp 588 599 the author proposed a new error and stepsize control strategy based on controlling the size of the defect of the underlying continuous solution that is particularly appropriate for numerical methods based on this class of formulas in this investigation the author presents an analysis of the relative efficiency of existing continuous runge kutta methods the author introduces theoretical measures useful in evaluating the potential of such methods and apply these measures to several methods of orders 4 through 8 the author also presents numerical results to illustrate that the theoretical measures of potential are consistent with the observed performance of nonstiff problems over a wide range of error tolerances

repairable consecutive k out of n f system with fuzzy states in this paper we study a repairable consecutive k out of n f system with fuzzy states the maintenance of the system is carried out by r repairmen or repair facilities the system is either circular or linear it is assumed that the working time and the repair time of each component in the system are both exponentially distributed and every component after repair is as same as new each component is classified as either a key component or an ordinary one according to its priority role to the system s repair when the system displays a gradual degradation of its performance its reliability is then analyzed in terms of fuzzy failure and success states the difference between fuzzy reliability theory and the conventional reliability theory is highlighted

fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context abstract conventional reliability theory is based on the binary state assumption and the probability assumption namely 1 the systems are precisely defined as functioning or failed 2 the system behavior is fully characterized in the context of probability measures however the probability assumption is no longer reasonable in the case of small size of samples since the term probability stems from the sample generality in this way possibility measures should be used to characterize the sample particularity in this paper the binary state assumption is reserved and the possibility assumption is taken in place of the probability assumption thus a concept of fuzzy reliability i e posbist reliability is introduced and a theory of fuzzy reliability i e posbist reliability theory is established various conclusions which are contrary to statements in conventional reliability theory i e probist reliability theory are then drawn

a general formal approach for fuzzy reliability analysis in the possibility context abstract in this paper we focus on the fuzzy reliability analysis in the posbist reliability theory in accordance with this theory the system failure state is defined precisely and the system failure and repair behaviour is fully characterized in the context of possibility measures to formalize the reliability analysis in the posbist theory we propose a general approach on the basis of a system of functional equations the solution of the system allows us to obtain fuzzy reliability characteristics of the arbitrary repairable and unrepairable systems to illustrate the construction and application of the system of functional equations various examples are given

dynamic programming for a markov switching jump diffusion abstract we consider an optimal control problem with a deterministic finite horizon and state variable dynamics given by a markov switching jump diffusion stochastic differential equation our main results extend the dynamic programming technique to this larger family of stochastic optimal control problems more specifically we provide a detailed proof of bellman s optimality principle or dynamic programming principle and obtain the corresponding hamilton jacobi belman equation which turns out to be a partial integro differential equation due to the extra terms arising from the levy process and the markov process as an application of our results we study a finite horizon consumption investment problem for a jump diffusion financial market consisting of one risk free asset and one risky asset whose coefficients are assumed to depend on the state of a continuous time finite state markov process we provide a detailed study of the optimal strategies for this problem for the economically relevant families of power utilities and logarithmic utilities

variation analysis of uncertain stationary independent increment processes a stationary independent increment process is an uncertain process with stationary and independent increments this paper aims to calculate the variance of stationary independent increment processes and gains that for each fixed time the variance is a constant multiplying the square of time based on this result it is proved that the total variation of stationary independent increment process with finite variance is bounded almost surely besides the quadratic variation of stationary independent increment process with finite variance is 0 almost surely and in mean

linear quadratic uncertain differential game with application to resource extraction problem uncertain differential game investigates interactive decision making of players over time and the system dynamics is described by an uncertain differential equation this paper goes further to study the two player zero sum uncertain differential game in order to guarantee the saddle point nash equilibrium a max min theorem is provided furthermore when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic the existence of saddle point nash equilibrium is obtained via the solvability of a corresponding riccati equation finally a resource extraction problem is analyzed by using the theory proposed in this paper