an efficient hybrid approach for multiobjective optimization of water distribution systems an efficient hybrid approach for the design of water distribution systems wdss with multiple objectives is described in this paper the objectives are the minimization of the network cost and maximization of the network resilience a self adaptive multiobjective differential evolution samode algorithm has been developed in which control parameters are automatically adapted by means of evolution instead of the presetting of fine tuned parameter values in the proposed method a graph algorithm is first used to decompose a looped wds into a shortest distance tree t or forest and chords the original two objective optimization problem is then approximated by a series of single objective optimization problems of the t to be solved by nonlinear programming nlp thereby providing an approximate pareto optimal front for the original whole network finally the solutions at the approximate front are used to seed the samode algorithm to find an improved front for the original entire network the proposed approach is compared with two other conventional full search optimization methods the samode algorithm and the nsga ii that seed the initial population with purely random solutions based on three case studies a benchmark network and two real world networks with multiple demand loading cases results show that i the proposed nlp samode method consistently generates better quality pareto fronts than the full search methods with significantly improved efficiency and ii the proposed samode algorithm no parameter tuning exhibits better performance than the nsga ii with calibrated parameter values in efficiently offering optimal fronts

markov processes with identical last exit distributions letx andy be two transient locally hunt markov processes ifx andy enjoy the same last exit distributions from compact sets theny is equivalent to a time change ofx by the inverse of a strictly increasing continuous additive functional this result can also be interpreted with natural auxiliary hypotheses as a statement in potential theory involving equilibrium measures

enhanced lower bounds for the global optimization of water distribution networks in this paper we address a global optimization approach to the problem of designing a water distribution network that satisfies specified flow demands at stated pressure head requirements the nonlinear nonconvex network problem is transformed into the space of certain design variables by relaxing the nonlinear constraints in the transformed space via suitable polyhedral outer approximations we derive a linear lower bounding problem this problem provides an enhancement of eiger et al s 1994 lower bounding scheme and takes advantage of the monotone concave convex nature of the nonlinear constraints upper bounds are computed by solving a projected linear program that uses the flow conserving solution generated by the lower bounding problem these bounding strategies are embedded within a branch and bound algorithm the partitioning scheme employed reduces the gap from optimality inducing a convergent process to a feasible solution that lies within any prescribed accuracy tolerance of global optimality the approach is illustrated using two standard test problems from the literature for the larger hanoi network problem a better incumbent than previously reported in the literature is obtained and is proven to be within 0 486 of optimality over the specified feasible domain used for this problem

global optimum tree solution for single source looped water distribution networks subjected to a single loading pattern a methodology based on linear programming lp is developed for obtaining the global optimum tree solution for single source looped water distribution networks wdns subjected to a single loading pattern initially a multiple link loop is considered and it is shown that the cost discharge curve is concavoconvex convex for local minima corresponding to trees and concave for discharges in between the logic is then extended to p loop networks for which the cost discharge surface is a p 1 dimensional hypersurface a tree of the looped wdn is initially selected optimized by lp and successively modified considering one loop at a time so that the solution jumps on the hypersurface from one local minimum to a better one the procedure is continued until no further improvement in local optimum solution occurs thus the final solution is the global optimum tree solution the optimization procedure is illustrated through a one source 31 demand node three loop wdn for which only 23 trees out of possible 1048 trees were required to be optimized for obtaining the global optimum tree solution since each link had to carry a minimum specified flow one more iteration was carried out in the end to satisfy the minimum flow constraint even though the tree solution is global optimum the final solution satisfying the minimum flow constraint cannot be claimed to be necessarily the global optimum

exchange leavitt path algebras and stable rank abstract we characterize those leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1 2 or concrete criteria in terms of properties of the underlying graph are given for each case

a combined nlp differential evolution algorithm approach for the optimization of looped water distribution systems 1 this paper proposes a novel optimization approach for the least cost design of looped water distribution systems wdss three distinct steps are involved in the proposed optimization approach in the first step the shortest distance tree within the looped network is identified using the dijkstra graph theory algorithm for which an extension is proposed to find the shortest distance tree for multisource wdss in the second step a nonlinear programming nlp solver is employed to optimize the pipe diameters for the shortest distance tree chords of the shortest distance tree are allocated the minimum allowable pipe sizes finally in the third step the original looped water network is optimized using a differential evolution de algorithm seeded with diameters in the proximity of the continuous pipe sizes obtained in step two as such the proposed optimization approach combines the traditional deterministic optimization technique of nlp with the emerging evolutionary algorithm de via the proposed network decomposition the proposed methodology has been tested on four looped wdss with the number of decision variables ranging from 21 to 454 results obtained show the proposed approach is able to find optimal solutions with significantly less computational effort than other optimization techniques

a modified linear programming gradient method for optimal design of looped water distribution networks a modified linear programming gradient lpg method is presented for solving looped water distribution network problems together with a mathematically rigorous derivation of the lpg model the lpg method of alperovits and shamir is modified in terms of both search direction and step size a quasi newton search direction is proposed instead of the steepest descent direction and the step size is determined by a backtracking line search method instead of a fixed step size the modified method is applied to a numerical example where it provides an improved solution in comparison to the original lpg method

an efficient constraint handling method for genetic algorithms many real world search and optimization problems involve inequality and or equality constraints and are thus posed as constrained optimization problems in trying to solve constrained optimization problems using genetic algorithms gas or classical optimization methods penalty function methods have been the most popular approach because of their simplicity and ease of implementation however since the penalty function approach is generic and applicable to any type of constraint linear or nonlinear their performance is not always satisfactory thus researchers have developed sophisticated penalty functions specific to the problem at hand and the search algorithm used for optimization however the most difficult aspect of the penalty function approach is to find appropriate penalty parameters needed to guide the search towards the constrained optimum in this paper ga s population based approach and ability to make pair wise comparison in tournament selection operator are exploited to devise a penalty function approach that does not require any penalty parameter careful comparisons among feasible and infeasible solutions are made so as to provide a search direction towards the feasible region once sufficient feasible solutions are found a niching method along with a controlled mutation operator is used to maintain diversity among feasible solutions this allows a real parameter ga s crossover operator to continuously find better feasible solutions gradually leading the search near the true optimum solution gas with this constraint handling approach have been tested on nine problems commonly used in the literature including an engineering design problem in all cases the proposed approach has been able to repeatedly find solutions closer to the true optimum solution than that reported earlier

a characterization of h brownian motion by its exit distributions letx h be anh brownian motion in the unit balld r d withh harmonic such that the representing measure ofh is not singular with respect to the surface measure on d ify is a continuous strong markov process ind with the same killing distributions asx h theny is a time change ofx h similar results hold in simply connected domains inc provided with either the martin or the euclidean boundary

the leavitt path algebra of a graph abstract for any row finite graph e and any field k we construct the leavitt path algebra l e having coefficients in k when k is the field of complex numbers then l e is the algebraic analog of the cuntz krieger algebra c e described in i raeburn graph algebras in cbms reg conf ser math vol 103 amer math soc 2005 the matrix rings m n k and the leavitt algebras l 1 n appear as algebras of the form l e for various graphs e in our main result we give necessary and sufficient conditions on e which imply that l e is simple

multiobjective design of water distribution systems under uncertainty 1 the water distribution system wds design problem is defined here as a multiobjective optimization problem under uncertainty the two objectives are 1 minimize the total wds design cost and 2 maximize wds robustness the wds robustness is defined as the probability of simultaneously satisfying minimum pressure head constraints at all nodes in the network decision variables are the alternative design options for each pipe in the network the sources of uncertainty are future water consumption and pipe roughness coefficients uncertain variables are modeled using probability density functions pdfs assigned in the problem formulation phase the corresponding pdfs of the analyzed nodal heads are calculated using the latin hypercube sampling technique the optimal design problem is solved using the newly developed rnsgaii method based on the nondominated sorting genetic algorithm ii nsgaii in rnsgaii a small number of samples are used for each fitness evaluation leading to significant computational savings when compared to the full sampling approach chromosome fitness is defined here in the same way as in the nsgaii optimization methodology the new methodology is tested on several cases all based on the new york tunnels reinforcement problem the results obtained demonstrate that the new methodology is capable of identifying robust pareto optimal solutions despite significantly reduced computational effort

optimisation of gravity fed water distribution network design a critical review abstract this paper surveys the literature on the optimisation of water distribution network design the water distribution network design wdnd optimisation problem entails finding the material and diameter of each pipe in the network so that the total cost of the network is minimised without violating any hydraulic constraints this is a difficult combinatorial optimisation problem in which decision variables are discrete and both cost function and constraints are non linear over the past 30 years a large number of methods especially in the field of meta heuristics have been developed to solve this problem most of which obtain good results on the available benchmark networks in addition to outlining the basic features of each method a detailed computational comparison is presented based on this comparison some issues with the current state of the art in this domain are discussed and some future research directions are suggested additionally the need for an adequate set of benchmark instances is motivated and the minimal requirements for an instance set generator are discussed

robust optimization for water distribution systems least cost design 1 the objective of the least cost design problem of a water distribution system is to find its minimum cost with discrete diameters as decision variables and hydraulic controls as constraints the goal of a robust least cost design is to find solutions which guarantee its feasibility independent of the data i e under model uncertainty a robust counterpart approach for linear uncertain problems is adopted in this study which represents the uncertain stochastic problem as its deterministic equivalent robustness is controlled by a single parameter providing a trade off between the probability of constraint violation and the objective cost two principal models are developed uncorrelated uncertainty model with implicit design reliability and correlated uncertainty model with explicit design reliability the models are tested on three example applications and compared for uncertainty in consumers demands the main contribution of this study is the inclusion of the ability to explicitly account for different correlations between water distribution system demand nodes in particular it is shown that including correlation information in the design phase has a substantial advantage in seeking more efficient robust solutions

aleatoric and epistemic uncertainty in groundwater flow and transport simulation 1 the characterization of aleatory hydrogeological parameter uncertainty has traditionally been accomplished using probability theory however when consideration is given to epistemic as well as aleatory uncertainty probability theory is not necessarily appropriate this is especially the case where expert opinion is regarded as a suitable source of information when experts opine upon the uncertainty of a parameter value both aleatoric and epistemic uncertainties are introduced and must be modeled appropriately a novel approach to expert provided parameter uncertainty characterization can be defined that bridges an historical gap between probability theory and fuzzy set theory herein a random set a generalization of a random variable is employed to formalize expert knowledge and fuzzy sets are used to propagate this uncertainty to model estimates of contaminant transport the resultant random set based concentration estimates are shown to be more general than the corresponding random variable estimates in some cases the random set based results are shown as upper and lower probabilities that bound the corresponding random variable s cumulative distribution function

fuzzy random reliability of structures based on fuzzy random variables abstract using fuzzy random variables as basic variables the relationship between fuzzy random variables and fuzzy random events is established in this paper formulas of measures of a fuzzy random event and its cut set are presented the fuzzy random reliability of structures is defined on the basis of fuzzy measure and fuzzy integral further by using fuzzy random variables the fuzzy random failure function of structures is defined and the fuzziness between structural reliable state and failure state is also analyzed moreover the fuzzy random limit state equation of structures is built the probability calculation method of structural reliability under the reliable level is given finally the calculation procedures of structural fuzzy random reliability is proposed

representing last exit potentials as potentials of measures let xtbe a transient hunt process having a potential density u x chosen in the usual way no duality hypotheses are assumed let k be a closed set with m sup t xt k almost surely for f a bounded borel function p x left f left x m right m 0 right int u left x y right left 1 k f left y right k left dy right f left z right v c left dy dz right right where is a measure on e and cis a measure on e e so that c e k e 0 if xt is simply a right process and k is closed p x left 1 k f left x m right m text 0 right int u left x y right 1 k f left y right k left dy right if xtis a right process and k e is closed in the ray topology of x then p x left f left x m right m 0 right int u left x y right left 1 k f left y right k left dy right f left z right v c left dy dz right right if xtis a diffusion we obtain the representation of equilibrium potentials for closed sets due to chung getoor sharpe and meyer without duality hypotheses

optimal design of water distribution networks optimal design of a water distribution network is formulated as a two stage decomposition model the master outer problem is nonsmooth and nonconvex while the inner problem is linear a semi infinite linear dual problem is presented and an equivalent finite linear problem is developed the overall design problem is solved globally by a branch and bound algorithm using nonsmooth optimization and duality theory the algorithm stops with a solution and a global bound such that the difference between this bound and the true global optimum is within a prescribed tolerance the algorithm has been programmed and applied to a number of examples from the literature the results demonstrate its superiority over previous methods

global minimization by reducing the duality gap we derive a general principle demonstrating that by partitioning the feasible set the duality gap existing between a nonconvex program and its lagrangian dual can be reduced and in important special cases even eliminated the principle can be implemented in a branch and bound algorithm which computes an approximate global solution and a corresponding lower bound on the global optimal value the algorithm involves decomposition and a nonsmooth local search numerical results for applying the algorithm to the pooling problem in oil refineries are given

distribution network optimization finding the most economic solution by using genetic algorithms abstract the problem of finding the optimal features in a distribution network is approached from an economic viewpoint this problem is closely related to drops in pressure within the network and is a complex mathematical problem with some empirically adjusted formulae that make the problem very difficult to solve analytically thus a method is proposed for finding the optimal networl features based on a well known optimization procedure genetic algorithms which have proven to be useful in this type of problems furthermore new problem specific genetic operators are presented and shown to be intuitively and in practice better than the standard ones

a two phase decomposition method for optimal design of looped water distribution networks a two phase decomposition method is proposed for the optimal design of new looped water distribution networks as well as for the parallel expansion of existing ones the main feature of the method is that it generates a sequence of improving local optimal solutions the first phase of the method takes a gradient approach with the flow distribution and pumping heads as decision variables and is an extension of the linear programming gradient method proposed by alperovits and shamir 1977 for nonlinear modeling the technique is iterative and produces a local optimal solution in the second phase the link head losses of this local optimal solution are fixed and the resulting concave program is solved for the link flows and pumping heads these then serve to restart the first phase to obtain an improved local optimal solution the whole procedure continues until no further improvement can be achieved some applications and extensions of the method are also discussed

a critical study of the linear programming gradient method for optimal design of water supply networks the linear programming gradient lpg method originally proposed by alperovits and shamir 1977 and the subsequent publications based on it are briefly reviewed the optimal solutions obtained by different investigators for a simple two loop network originally taken by alperovits and shamir as an illustrative example are compared and it is shown that they are not even the local optimum solutions the latest modification introduced by kessler and shamir 1989 which incorporates the projected gradient of the objective function is also briefly reviewed and is shown to lead to a truly local optimum solution as claimed by kessler and shamir it is shown that it is preferable initially to suppress the minimum link flow constraint necessary for reliability purposes and introduce it later after obtaining the optimal solution for a branching configuration the philosophy of the lpg method is analyzed it is shown that the lpg method is inefficient at least for optimization of the illustrative two loop network as compared to a heuristic method that initially identifies logically good branching configurations for obtaining the optimal solution

algebras of quotients of path algebras abstract leavitt path algebras are shown to be algebras of right quotients of their corresponding path algebras using this fact we obtain maximal algebras of right quotients from those leavitt path algebras whose associated graph satisfies that every vertex connects to a line point equivalently the leavitt path algebra has essential socle we also introduce and characterize the algebraic counterpart of toeplitz algebras

risk model with fuzzy random individual claim amount in this paper we consider a risk model in which individual claim amount is assumed to be a fuzzy random variable and the claim number process is characterized as a poisson process the mean chance of the ultimate ruin is researched particularly the expressions of the mean chance of the ultimate ruin are obtained for zero initial surplus and arbitrary initial surplus if individual claim amount is an exponentially distributed fuzzy random variable the results obtained in this paper coincide with those in stochastic case when the fuzzy random variables degenerate to random variables finally two numerical examples are presented

rings of quotients of incidence algebras and path algebras we compute the maximal right left symmetric rings of quotients of finite dimensional incidence and graph algebras we show that these rings of quotients are morita equivalent to incidence algebras and path algebras respectively with respect to simpler well determined partially ordered sets and finite quivers respectively

quantum spheres and projective spaces as graph algebras the c algebras of continuous functions on quantum spheres quantum real projective spaces and quantum complex projective spaces are realized as cuntz krieger algebras corresponding to suitable directed graphs structural results about these quantum spaces especially about their ideals and k theory are then derived from the general theory of graph algebras it is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle respectively

non simple purely infinite rings in this paper we introduce the concept of purely infinite rings which in the simple case agrees with the already existing notion of pure infiniteness we establish various permanence proper ties of this notion with respect to passage to matrix rings corners and behaviour under extensions so being purely infinite is preserved under morita equivalence we show that a wealth of examples falls into this class including important analogues of constructions commonly found in operator al gebras in particular for any s unital k algebra having enough nonzero idempotents for example for a von neumann regular algebra its tensor product over k with many non simple leavitt path algebras is purely infinite introduction the notion of pure infiniteness has proved key in the the ory of operator algebras since its conception in the early eighties by j cuntz see 27 this was done for simple algebras and provided a huge list of exam ples one of the milestones of the theory became the classification of separable nuclear unital purely infinite simple algebras by means of k theoretic invariants due to kirchberg 41 and phillips 58

reliability and mean time to failure of unrepairable systems with fuzzy random lifetimes in practice the lifetimes of unrepairable systems contain both randomness and fuzziness so it is appropriate to assume the lifetimes of systems to be fuzzy random variables in this paper the definitions of reliability and mean time to failure mttf of unrepairable systems with fuzzy random lifetimes are given respectively then basic mathematical models of unrepairable systems with fuzzy random lifetimes are established furthermore the reliability and mttf of series systems parallel systems series parallel systems parallel series systems and cold standby systems are discussed respectively finally some numerical examples are presented to illustrate how to calculate the reliability and mttf of unrepairable systems with fuzzy random lifetimes

uniqueness theorems and ideal structure for leavitt path algebras abstract we prove leavitt path algebra versions of the two uniqueness theorems of graph c algebras we use these uniqueness theorems to analyze the ideal structure of leavitt path algebras and give necessary and sufficient conditions for their simplicity we also use these results to give a proof of the fact that for any graph e the leavitt path algebra l c e embeds as a dense subalgebra of the graph c algebra c e this embedding has consequences for graph c algebras and we discuss how we obtain new information concerning the construction of c e

fuzzy probabilistic design of water distribution networks 1 the primary aim of this paper is to present a fuzzy probabilistic approach for optimal design and rehabilitation of water distribution systems combining aleatoric and epistemic uncertainties in a unified framework the randomness and imprecision in future water consumption are characterized using fuzzy random variables whose realizations are not real but fuzzy numbers and the nodal head requirements are represented by fuzzy sets reflecting the imprecision in customers requirements the optimal design problem is formulated as a two objective optimization problem with minimization of total design cost and maximization of system performance as objectives the system performance is measured by the fuzzy random reliability defined as the probability that the fuzzy head requirements are satisfied across all network nodes the satisfactory degree is represented by necessity measure or belief measure in the sense of the dempster shafer theory of evidence an efficient algorithm is proposed within a monte carlo procedure to calculate the fuzzy random system reliability and is effectively combined with the nondominated sorting genetic algorithm ii nsgaii to derive the pareto optimal design solutions the newly proposed methodology is demonstrated with two case studies the new york tunnels network and hanoi network the results from both cases indicate that the new methodology can effectively accommodate and handle various aleatoric and epistemic uncertainty sources arising from the design process and can provide optimal design solutions that are not only cost effective but also have higher reliability to cope with severe future uncertainties

structure of leavitt path algebras of polynomial growth we determine the structure of leavitt path algebras of polynomial growth and discuss their automorphisms and involutions

imprecise probabilistic evaluation of sewer flooding in urban drainage systems using random set theory 1 uncertainty analysis is widely applied in water system modeling to quantify prediction uncertainty from models and data conventional methods typically handle various kinds of uncertainty using a single characterizing approach be it probability theory or fuzzy set theory however using a single approach may not be appropriate particularly when uncertainties are of different types for example in sewer flood estimation problems random rainfall variables are used as model inputs and imprecise or subjective information is used to define model parameters this paper presents a general framework for sewer flood estimation that enables simultaneous consideration of two types of uncertainty randomness from rainfall data represented using imprecise probabilities and imprecision from model parameters represented by fuzzy numbers these two types of uncertainties are combined using random set theory and then propagated through a hydrodynamic urban drainage model two propagation methods i e discretization and monte carlo based methods are presented and compared with the latter shown to be much more computationally efficient and hence recommended for high dimensional problems the model output flood depth is generated in the form of lower and upper cumulative probabilities which are best estimates given the various stochastic and epistemic uncertainties considered and which embrace the unknown true cumulative probability the distance between the cumulative probabilities represents the extent of imprecise incomplete or conflicting information and can be reduced only when more knowledge is available the proposed methodology has a more complete and thus more accurate representation of uncertainty in data and models and can effectively handle different uncertainty characterizations in a single integrated framework for sewer flood estimation

tabu search algorithms for water network optimization abstract in this paper we propose a tabu search algorithm to find the least cost design of looped water distribution networks the mathematical nature of this optimization problem a nonlinear mixed integer problem is at the origin of a multitude of contributions to the literature in the last 25 years in fact exact optimization methods have not been found for this type of problem and in the past classical optimization methods like linear and nonlinear programming were tried at the cost of drastic simplifications tabu search is a valuable heuristic technique for solving problems cast in combinatorial form this is based on the human memory process and uses an iterative neighborhood search procedure in an attempt to avoid becoming trapped in local optima the use of such a heuristic procedure to solve the aforementioned problem needs particular tailoring to produce high quality solutions in this paper we present the essential features of the algorithm and the results obtained when it is applied to some of the classical water distribution network case studies appearing in the literature the results are very promising and demonstrate the usefulness of tabu search algorithms in solving this kind of optimization problem

a diameter sensitive flow entropy method for reliability consideration in water distribution system design flow entropy is a measure of uniformity of pipe flows in water distribution systems by maximizing flow entropy one can identify reliable layouts or connectivity in networks in order to overcome the disadvantage of the common definition of flow entropy that does not consider the impact of pipe diameter on reliability an extended definition of flow entropy termed as diameter sensitive flow entropy is proposed this new methodology is then assessed by using other reliability methods including monte carlo simulation a pipe failure probability model and a surrogate measure resilience index integrated with water demand and pipe failure uncertainty the reliability assessment is based on a sample of wds designs derived from an optimization process for each of the two benchmark networks correlation analysis is used to evaluate quantitatively the relationship between entropy and reliability to ensure reliability a comparative analysis between the flow entropy and the new method is conducted the results demonstrate that the diameter sensitive flow entropy shows consistently much stronger correlation with the three reliability measures than simple flow entropy therefore the new flow entropy method can be taken as a better surrogate measure for reliability and could be potentially integrated into the optimal design problem of wdss sensitivity analysis results show that the velocity parameters used in the new flow entropy has no significant impact on the relationship between diameter sensitive flow entropy and reliability

global optimization of nonlinear network design a novel approach for obtaining globally optimal solutions to design of networks with nonlinear resistances and potential driven flows is proposed the approach is applicable to networks where the potential loss on an edge in the network is governed by a convex and strictly monotonically increasing function of flow rate we introduce a relaxation of the potential loss constraint and formulate the design problem as a mixed integer nonlinear program a linearization based approach with tailored cuts is proposed that improves the computational efficiency over a standard implementation we have also implemented a simple heuristic approach for finding feasible solutions at the root node and during the search process the algorithm has been implemented with ibm ilog cplex and is shown to be computationally effective on a number of examples from the literature

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

markov processes with identical hitting probabilities let x t px and y t qx be transient hunt processes on a state space e satisfying the hypothesis of absolute continuity meyer s hypothesis l let t k be the first entrance time into a set k and assume px t k t then x t t px and y t qx have the same joint distributions an analogous result is stated if x and y are right processes with an additional hypothesis these theorems generalize the blumenthal getoor mckean theorem and have interpretations in terms of potential theory 0 introduction in 1962 blumenthal getoor and mckean 2 proved the following theorem in a paper entitled markov processes with identical hitting distributions 0 1 theorem let xt px and yt qx be standard processes on ea s so that for each compact subset k of e pk x qk x for all x there exists a continuous additive functional at of xt which is strictly increasing and finite on 0 so that if tt is the right continuous inverse of at then x tt px and y qx have the same joint distributions here pk x and qk x are the distributions of x and y at tk the first hitting time of k their proof works equally well if x and y are right processes the hypotheses droites were formulated later than 1962 now this theorem is of interest not only in probability theory but in potential theory as well the object of interest in potential theory is the cone of excessive functions associated to the markov process hunt s balayage theorem 1 111 6 12 tells how to construct the hitting operators pk given the cone of excessive functions and dynkin s theorem 1 11 5 1 tells how to determine the cone of excessive furnctions given the hitting operators thus theorem 0 1 states that if two right processes have the same potential theory i e the same excessive functions then they are related by a time change as described above also it is simple to observe that time changing x in the fashion described in 0 1 leaves the hitting operators unchanged so the potential theory does not change received by the editors october 7 1981 1980 mathematics subject classification primary 60j25 secondary 31c15