hardness of solving relational equations minimal solutions play a crucial role in describing all solutions of relational equations for this reason the problem of computing minimal solutions has for long been examined the literature contains several algorithms for computing minimal solutions recently contributions regarding computational complexity of the problem itself appeared the complexity aspect is clearly of fundamental importance however the existing results contain serious flaws in this paper we inspect the existing contributions clarify the flaws examine the problem of complexity of computing minimal solutions prove that there is no efficient algorithm computing all minimal solutions and discuss further ramifications of our observations

on the minimal solutions of max min fuzzy relational equations in this paper the minimal solutions of max min fuzzy relational equations are investigated a sufficient and necessary condition for discriminating whether a given solution is minimal or not is shown furthermore we propose a new algorithm for computing all minimal solutions less than or equal to a given one

resolution of a system of the max product fuzzy relation equations using lou factorization in this paper the lu factorization is extended to the fuzzy square matrix with respect to the max product composition operator called l u factorization equivalently we will find two fuzzy lower and upper triangular matrices l and u for a fuzzy square matrix a such that a l u where is the max product composition an algorithm is presented to find the matrices l and u furthermore some necessary and sufficient conditions are proposed for the existence and uniqueness of the l u factorization for a given fuzzy square matrix a an algorithm is also proposed to find the solution set of a square system of fuzzy relation equations fre using the l u factorization the algorithm finds the solution set without finding its minimal solutions and maximum solution it is shown that the two algorithms have a polynomial time complexity as o n 3 since the determination of the minimal solutions is an np hard problem the algorithm can be very important from the practical point of view

fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max min composition in this paper we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max min composition in which fuzzy inequality replaces ordinary inequality in the constraints fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with ordinary inequality constraints denoted by x o i in this paper we have presented an algorithm to generate such optimal solutions also two examples are given to illustrate the algorithm presented and some real application of this work

the oseen drag on a sphere and the method of induced forces abstract the method of induced forces which was used previously to derive a scheme for the evaluation of the oseen drag exerted on a circular cylinder is applied now to the problem of the oseen drag experienced by a sphere by taking into account the zeroth and first force multipole alone very satisfactory values for the drag coefficient are obtained for reynolds numbers up to order 10 2

deriving minimal solutions for fuzzy relation equations with max product composition this work considers fuzzy relation equations with max product composition the critical problem in solving such equations is to determine the minimal solutions when an equation is solvable however this problem is np hard and difficult to solve a v markovskii on the relation between equations with max product composition and the covering problem fuzzy sets and systems 153 2005 261 273 this work first examines the attributes of a solvable equation and characteristics of minimal solutions then reduces the equation to an irreducible form and converts the problem into a covering problem for which minimal solutions are correspondingly determined furthermore for theoretical and practical applications this work presents a novel method for obtaining minimal solutions the proposed method easily derives a minimal solution and obtains other minimal solutions from this predecessor using a back tracking step the proposed method is compared with an existing algorithm and some applications are described in detail

oscillations in the coefficients of power series the euler transformation van dyke quart j mech app math 27 1974 pp 423 450 pearce adv in phys 27 1978 pp 89 148 is a standard tool of series analysis for extending the range of convergence of a numerically determined series expansion in powers of some essentially positive physical parameter r because it can be used to map the physically significant range 0 r infty inside the unit circle the transformed series may in principle be useful for all positive r and for studying asymptotic behavior as r to infty in practice the deduction of this limiting behavior from a finite number of the coefficients of the transformed series is often complicated by substantial and slowly decaying oscillations in these coefficients we show that such oscillations can arise when r infty is a nonisolated or essential singularity we prove a generalization of darboux s theorem that applies to this case but show that the oscillations are due to components of the function that is

solving a linear programming problem with the convex combination of the max min and the max average fuzzy relation equations abstract in this paper we introduce a fuzzy operator constructed by the convex combination of two known operators max min and max average compositions h j zimmermann fuzzy set theory and it s application kluwer academic publishers boston dordrecht london 1999 this operator contains some properties of the two known compositions when it generates the feasible region for linear optimization problems we investigate linear optimization problems whose feasible region is the fuzzy sets defined with this operator thus firstly the structure of these fuzzy regions is considered and then a method to solve the linear optimization problems with fuzzy equation constraints regarding this operator is presented

linear optimization with bipolar max min constraints we consider a generalization of the linear optimization problem with fuzzy relational in equality constraints by allowing for bipolar max min constraints i e constraints in which not only the independent variables but also their negations occur a necessary condition to have a non empty feasible domain is given the feasible domain if not empty is algebraically characterized a simple procedure is described to generate all maximizers of the linear optimization problem considered and is applied to various illustrative example problems

increasing continuous operations in fuzzy max equations and inequalities abstract in this paper solutions of fuzzy systems of equations a x b and systems of inequalities a x b a x b are considered where a 0 1 m n b 0 1 m stands for max product and x 0 1 n is an unknown vector in this paper the families of all solutions are described during the last thirty years this problem was considered by many authors our goal is to achieve the most general assumptions for which sets of solutions systems of equations and inequalities are the union of lattice intervals especially a method of determination of all minimal solutions in such systems is indicated as a result the families of all solutions of fuzzy systems of inequalities a x b and a x b are described as the main part of these results the method of determination of the family of all minimal solutions in max system of equations with an increasing operation continuous on the second argument is presented

latticized linear programming and fuzzy relation inequalities abstract a logical linear programming problem the latticized linear programming is proposed based on fuzzy lattice and fuzzy relation inequalities the proposed problem is essentially an optimization problem which should be useful under certain logical if then situations to illustrate the proposed approach numerical examples are solved

minimizing a linear function under a fuzzy max min relational equation constraint in this paper we investigate the problem of minimizing a linear objective function subject to a fuzzy relational equation constraint a necessary condition for optimal solution is proposed based on this necessary condition we propose three rules to simplify the work of computing an optimal solution numerical examples are provided to illustrate the procedure experimental results are reported showing that our new procedure systematically outperforms our previous work

solving fuzzy relation equations with a linear objective function abstract an optimization model with a linear objective function subject to a system of fuzzy relation equations is presented due to the non convexity of its feasible domain defined by fuzzy relation equations designing an efficient solution procedure for solving such problems is not a trivial job in this paper we first characterize the feasible domain and then convert the problem to an equivalent problem involving 0 1 integer programming with a branch and bound solution technique after presenting our solution procedure a concrete example is included for illustration purpose

min max programming problem subject to addition min fuzzy relation inequalities a bittorrent like peer to peer file sharing system can be reduced into a system of addition min fuzzy relation inequalities addition min is a new composition and the solution set of addition min fuzzy relation inequalities differs much from that of general max t norm fuzzy relation inequalities or equations in order to avoid network congestion and improve the stability of data transmission a min max programming problem is proposed subject to addition min fuzzy relation inequalities in this paper based on some relevant theorems on resolution a novel algorithm is developed step by step to find an optimal solution of the proposed problem moreover an application example is given to illustrate the feasibility and efficiency of the algorithm finally some further discussions are made concerning the optimal solution of the proposed problem

latticized linear programming subject to max product fuzzy relation inequalities with application in wireless communication in this paper we introduce the latticized linear programming problem subject to max product fuzzy relation inequalities with application in the optimization management model of wireless communication emission base stations resolution of max product fuzzy relation inequalities is studied by comparing with that of the corresponding max product fuzzy relation equations a solution matrix approach is developed for solving the proposed problem without finding all the quasi minimal solutions of the constraint for carrying out the solution matrix approach we provide a step by step algorithm illustrated by a numerical example

optimization of fuzzy relational equations with max av composition max min and max product compositions are commonly utilized to optimize a linear objective function subject to fuzzy relational equations both are members in the class of max t norm composition in this study the max av composition is considered for the same optimization model which does not belong to the max t norm composition however max av composition generates some properties of the solution set that are similar to the max product composition thanks to these properties a simple value matrix with rules can be applied to reduce problem size thus this study proposes an efficient procedure for obtaining optimal solutions without decomposing the problem into two sub problems or finding all the potential minimal solutions

the analytic structure of oseen flow past a sphere as a function of reynolds number the feasibility of van dyke s 28 proposal that one go from zero to infinite reynolds number r by computer extension of a stokes series in powers of r depends on the analytic structure of the flow as a function of r this structure is investigated in detail for oseen flow past a sphere we find that there is a doubly infinite array of simple pole singularities in the left half complex r plane each of which lies close to one of the n zeros of the modified bessel functions k n 1 2 r for some positive integer n consequently r infty is an accumulation point of poles and hence a nonisolated singularity we have extended the stokes series for the drag coefficient c d r to 66 terms in double precision and have sought the asymptotic behavior of c d r as r to infty neither pade approximants nor an euler transformation formed from our long series give good convergence we suggest that the asymptotic behavior of c d r is primarily as an expansion in powers of r 2 3 though

an accelerated approach for solving fuzzy relation equations with a linear objective function in literature the optimization model with a linear objective function subject to fuzzy relation equations has been converted into a 0 1 integer programming problem by fang and li 1999 they proposed a jump tracking branch and bound method to solve this 0 1 integer programming problem in this paper we propose an upper bound for the optimal objective value based on this upper bound and rearranging the structure of the problem we present a backward jump tracking branch and bound scheme for solving this optimization problem a numerical example is provided to illustrate our scheme furthermore testing examples show that the performance of our scheme is superior to the procedure in the paper by fang and li several testing examples show that our initial upper bound is sharp

minimal solutions of generalized fuzzy relational equations probabilistic algorithm based on greedy approach abstract the paper deals with generalized fuzzy relational equations that are defined within a recently introduced framework of sup preserving aggregation structures generalized fuzzy relational equations subsume all previously studied types of fuzzy relational equations namely those that are based on sup t norm and inf residuum compositions the paper contributes to previous studies of generalized fuzzy relational equations by presenting a method for constructing all minimal solutions and consequently for determining the whole solution set for any given generalized fuzzy relational equation that is solvable and for which every solution is bounded from below by a minimal solution moreover in the paper we present a simple probabilistic algorithm for finding all minimal solutions

fuzzy relational equations on complete brouwerian lattices this paper deals with the problem of solving fuzzy relational equations with inf a t composition on complete brouwerian lattices under finite domains resp countably infinite domains when the right hand sides of fuzzy relational equations are meet irreducible elements or have irredundant finite meet decompositions some necessary and sufficient conditions for attainable solutions resp unattainable solutions are formulated and some properties of the attainable solutions resp the unattainable solutions are shown further the solution sets are given when they are nonempty

on the oseen drag on a sphere the method of induced forces is employed to evaluate the oseen drag on a sphere the accuracy of the values obtained for the drag coefficient for large reynolds numbers is increased by application of shanks s transformation on the basis of the improved values we obtain as numerical approximations for the exponent of the first term in the asymptotic expansion values approaching 0 660 for reynolds numbers between 10 2 and 10 6

computing improved optimal solutions to max min flexible constraint satisfaction problems abstract the formal framework for decision making in a fuzzy environment is based on a general max min bottleneck like optimization problem proposed by zadeh it is also the basis for extending the constraint satisfaction paradigm of artificial intelligence to accommodating flexible or prioritized constraints this paper surveys refinements of the ordering of solutions supplied by the max min formulation namely the discrimin partial ordering and the leximin complete preordering a general algorithm is given which computes all maximal solutions in the sense of these relations it also sheds light on the structure of the set of best solutions moreover classes of problems for which there is a unique best discrimin and leximin solution are exhibited namely continuous problems with convex domains and so called isotonic problems noticeable examples of such problems are fuzzy linear programming problems and fuzzy pert like scheduling problems

computing leximin optimal solutions in constraint networks in many real world multiobjective optimization problems one needs to find solutions or alternatives that provide a fair compromise between different conflicting objective functions which could be criteria in a multicriteria context or agent utilities in a multiagent context while being efficient i e informally ensuring the greatest possible overall agents satisfaction this is typically the case in problems implying human agents where fairness and efficiency requirements must be met preference handling resource allocation problems are another examples of the need for balanced compromises between several conflicting objectives a way to characterize good solutions in such problems is to use the leximin preorder to compare the vectors of objective values and to select the solutions which maximize this preorder in this article we describe five algorithms for finding leximin optimal solutions using constraint programming three of these algorithms are original other ones are adapted in constraint programming settings from existing works the algorithms are compared experimentally on three benchmark problems

refinements of the maximin approach to decision making in a fuzzy environment abstract the most popular approach to decision making in the setting of fuzzy sets is the maximin ranking of solutions this method is natural when interpreting the fuzzy sets as flexible constraints that cannot compensate with one another however the obtained ranking of solutions is very coarse two kinds of refinements to this ordering are introduced a partial ordering according to the least satisfied discriminating constraint and a lexicographical ranking the latter refines the former and combines utilitarist and egalitarist points of view on the aggregation of feasibility degrees these orderings are characterized in several ways and their representation by means of two place numerical functions is studied dual refinements of the maximax ranking are provided

an algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition min composition abstract in this paper we study the optimal solution of minimizing a linear objective function subject to the fuzzy relation inequalities with addition min composition we first discuss some properties about the minimal solutions of fuzzy relation inequalities with addition min composition and define the pseudo minimal indexes of this system next we give an algorithm to get the set of the pseudo minimal indexes which is called pmi algorithm finally we obtain an algorithm for this optimization system by utilizing these concepts and results the example is provided to show that our algorithm is simple and convenient

addition min fuzzy relation inequalities with application in bittorrent like peer to peer file sharing system abstract the data transmission mechanism in bittorrent like peer to peer p2p file sharing systems may be reduced to some addition min fuzzy relation inequalities the solution set of addition min fuzzy relation inequalities plays an important role in the corresponding optimization problem in this paper we study some properties of the solutions to such system convexity of the solution set and number of minimal solutions are discussed with comparison to those of the classical max t fuzzy relation equations or inequalities besides vertex solution and variable ordering minimal solution are also investigated with application in bittorrent like p2p file sharing system two numerical examples are given to illustrate the feasibility and efficiency of the algorithm for solving the variable ordering solution

leximax minimum solution of addition min fuzzy relation inequalities abstract addition min fuzzy relation inequalities are introduced to characterize the peer to peer p2p network system in order to decrease the network congestion considering both efficiency and fairness we define the leximax minimum solution for system of addition min fuzzy relation inequalities the leximax optimality refines both pareto optimality and max optimality for solving the leximax minimum solution concepts of minimum zero point and minimax value are defined with corresponding resolution algorithms i e algorithms i and ii based on these two algorithms algorithm iii is further developed to find the unique leximax minimum solution numerical examples illustrate the validity of our proposed algorithms

generalized analytic functions in axially symmetric oseen flows a class of generalized analytic functions arising from the oseen equations in the axially symmetric case has been identified for this class of functions the generalized cauchy integral formula has been obtained and a series representation for the region exterior to a sphere has been constructed the velocity field of the axially symmetric oseen flow has been represented in terms of two generalized analytic functions and it has been shown that for an exterior oseen flow problem those functions are uniquely determined provided that they both vanish at infinity also the pressure and vorticity have been determined as the real and imaginary parts of the two functions representing the velocity field and the drag exerted on a solid body of revolution in the axially symmetric oseen flow has been expressed in terms of one of the involved generalized analytic functions the problem of the axially symmetric oseen flow past a solid body of revolution has been reduced to an integral equation based on the gener

on fuzzy relational equations and the covering problem previous studies have shown that fuzzy relational equations fres based on either the max continuous archimedean t norm or the max arithmetic mean composition can be transformed into the covering problem which is an np hard problem exploiting the properties common to the continuous archimedean t norm and the arithmetic mean this study proposes a generalization of them as the u norm enabling fres that are based on the max continuous u norm composition also to be transformed into the covering problem this study also proposes a procedure for transforming the covering problem into max product fres consequently max continuous u norm fres can be solved by extending any procedure for solving either the covering problem or max product fres

an efficient procedure for solving a fuzzy relational equation with max archimedean t norm composition in the literature a necessary condition for minimal solutions of a fuzzy relational equation with max product composition shows that each of its components is either zero or the corresponding component s value of the greatest solution in this paper we first extend this necessary condition to the situation with max archimedean triangular norm t norm composition based on this necessary condition we then propose rules to reduce the problem size so that the complete set of minimal solutions can be computed efficiently furthermore rather than work with the actual equations we employ a simple matrix whose elements capture all of the properties of the equations in finding the minimal solutions numerical examples with specific cases of the max archimedean t norm composition are provided to illustrate the procedure

solutions and strong solutions of min product fuzzy relation inequalities with application in supply chain abstract min product fuzzy relation inequalities are introduced to describe the price requirements in the supply chain system we first investigate the resolution method and structure of the complete solution set to min product fuzzy relation inequalities motivated by the practical application we further define concept of strong solution a strong solution represents a feasible pricing scheduling enabling all the suppliers and retailers profitable in the trade activity relevant results about the strong solution are obtained based on such results a step by step algorithm is developed for finding all the strong solutions to a system of min product fuzzy relation inequalities numerical examples are given to illustrate the algorithm

decidable metric logics the common metric temporal logic for continuous time were shown to be insufficient when it was proved that they cannot express a modality suggested by pnueli moreover no finite temporal logic can express all the natural generalizations of this modality it followed that if we look for an optimal decidable metric logic we must accept infinitely many modalities or adopt a different formalism here we identify a fragment of the second order monadic logic of order with the 1 function that expresses all the pnueli modalities and much more its main advantage over the temporal logics is that it enables us to say not just that within prescribed time there is a point where some punctual event will occur but also that within prescribed time some process that starts now or that started before or that will start soon will terminate we prove that this logic is decidable with respect to satisfiability and validity over continuous time the proof depends heavily on the theory of compositionality in particular every temporal logic that has truth tables in this logic is automatically decidable we extend this result by proving that any temporal logic that has all its modalities defined by means more general than truth tables in a logic stronger than the one just described has a decidable satisfiability problem we suggest that this monadic logic can be the framework in which temporal logics can be safely defined with the guarantee that their satisfiability problem is decidable

on the relation between equations with max product composition and the covering problem systems of equations with max product composition are considered it is shown that solving these equations is closely related with the covering problem which belongs to the category of np hard problems it is proved that minimal solutions of equations correspond to irredundant coverings in terms of the covering problem the conditions of compatibility of equations of redundancy of equations of uniqueness of solution of uniqueness of minimal solution are determined concepts of essential non essential semi essential and super essential variables are suggested ways of simplification of a covering problem and methods of its solving are considered

lexicographic optimal solution of the multi objective programming problem subject to max product fuzzy relation inequalities abstract it is shown in this paper that the emission base stations in wireless communication can be reduced into a system of fuzzy relation inequalities with max product composition for optimal management in such system we introduce the fuzzy relation multi objective programming concept of feasible index set fis is defined based on which a novel algorithm named fis algorithm is developed to find the unique lexicographic optimal solution of the proposed problem with polynomial computational complexity applying this method we needn t to find out all the minimal solutions of the constraint a numerical application example is provided to illustrate the feasibility and efficiency of the fis algorithm

a linear programming approach for minimizing a linear function subject to fuzzy relational inequalities with addition min composition in this paper we study an optimization problem of minimizing a linear function subject to fuzzy relational inequalities with the addition min composition this optimization setting has recently been proposed to model the network cogestion issue when a bittorrent like peer to peer file sharing system is used for data transmission in a 2014 paper a pseudominimal index pmi based approach was proposed to search for an optimal solution it turns out that the pmi based approach may require to solve several to many linear programming problems in order to get an optimal solution in this paper we point out that the feasible domain is indeed convex and we only need to solve a single linear programming problem to generate an optimal solution for the original optimization problem furthermore our approach could be extended to the case with a nonlinear continuous objective function

timer formulas and decidable metric temporal logic we define a quantitative temporal logic that is based on a simple modality within the framework of monadic predicate logic its canonical model is the real line and not an w sequence of some type it can be interpreted either by behaviors with finite variability or by unrestricted behaviors for finite variability models it is as expressive as any logic suggested in the literature for unrestricted behaviors our treatment is new in both cases we prove decidability and complexity bounds using general theorems from logic and not from automata theory the technical proof uses a sublanguage of the metric monadic logic of order the language of timer normal form formulas metric formulas are reduced to timer normal form and timer normal form formulas allow elimination of the metric

a modified pso algorithm for linear optimization problem subject to the generalized fuzzy relational inequalities with fuzzy constraints fri fc in this paper optimization of a linear objective function subject to a generalized fuzzy relational inequalities is investigated in which an arbitrary continuous t norm is considered as fuzzy composition and fuzzy inequality replaces ordinary inequality in the constraints unlike most optimization algorithms in fuzzy relational inequalities with fuzzy constraints fri fc we find a near feasible solution having a better objective value than those resulting from the resolution of the similar problems with crisp ordinary inequality constraints such solutions are called super optima in this paper subsequently an algorithm is proposed to find a super optimum with pre specified desirable infeasibility for this purpose we firstly study some structural properties of the fri fc problem and present a new formulization that is independent of t norms used in the constraints of the problem this new formulation converts the primary problem into an equivalent problem with simple constraints without considering any penalty parameters however it is proved that the transformed equivalent problem enables our algorithm to distinguish unfavorable points and generate a sequence of solutions converging to a super optimum under a certain sufficient condition finally a modified pso is presented in which the ability of pso to solve unconstrained problems with continuous domain the structure of the transformed problem and some fuzzy structural modifications are combined and form an efficient algorithm to solve the generalized fri fc problems the modified pso algorithm has been applied to the generalized fri fc problem defined with ten well known continuous t norms additionally an idea of the fri fc problems as outer approximators for fri problems with ordinary inequalities is also investigated

optimal vector based algorithm for solving min max programming subject to addition min fuzzy relation inequality optimal management model in bittorrent like peer to peer p2p file sharing system can be reduced into a min max programming problem subject to addition min fuzzy relation inequality to deal with such problem we introduce the concept of optimal vector besides a novel optimal vector based ovb algorithm is developed to find a minimal optimal solution based on practical application consideration solution obtained by the ovb algorithm is theoretically demonstrated to be better than or equal to that obtained by the existing method advantage of our proposed ovb algorithm is illustrated by detailed numerical examples

power series expansions for the frequency and period of the limit cycle of the van der pol equation a power series expansion in the damping coefficient a is developed for the frequency nu varepsilon of the limit cycle of the van der pol equation ddot u u varepsilon dot u 1 u 2 0 leqq varepsilon infty the expansion is computed to o varepsilon 24 in rational arithmetic using the macsyma symbolic manipulation system and to o varepsilon 164 in floating point arithmetic using fortran a pade analysis of the power series expansion indicates that the singularities in the complex varepsilon 2 plane which are nearest the origin are branch points at radius r approx 3 42 and modulus pm beta where beta approx 1 radians introduction of the variable w varepsilon 2 varepsilon 4 2 varepsilon 2 r cos beta r 2 1 2 leads to expansions for the frequency and period which converge for 0 leqq w 1 i e for all varepsilon the results compare very favorably with published frequencies determined numerically but disagree to some exte