single variable term semi latticized fuzzy relation geometric programming with max product operator considering the practical application background in peer to peer network system we investigate the single variable term semi latticized fuzzy relation geometric programming with max product operator in this paper based on the characteristics of the objective function and feasible domain we develop a matrix approach to deal with the proposed problem a step by step algorithm is proposed to find an optimal solution without solving all the minimal solutions of the constraint two examples are given to illustrate the feasibility and efficiency of the algorithm

invariant manifolds of partial functional differential equations this paper is concerned with the existence smoothness and attractivity of invariant manifolds for evolutionary processes on general banach spaces when the nonlinear perturbation has a small global lipschitz constant and locally ck smooth near the trivial solution such a nonlinear perturbation arises in many applications through the usual cut off procedure but the requirement in the existing literature that the nonlinear perturbation is globally ck smooth and has a globally small lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut off function our general results are illustrated by and applied to partial functional differential equations for which the phase space c r 0 x where r 0 and x being a banach space has no smooth inner product structure and for which the validity of variation of constants formula is still an interesting open problem

the more for less paradox in fuzzy posynomial geometric programming the more for less mfl problem in fuzzy posynomial geometric programming fpgp is advanced in this paper the research results presented here focus primarily on the nonconvex fpgp in both objective functions and constraint functions convexification quasiconvex or pseudoconvex is extended in the sense of an mfl paradox by consolidating the necessary and sufficient conditions since the fpgp is equivalent to fuzzy linear programming correspondingly there exists a solution to the fpgp furthermore the duality or strong duality theorem the equivalent condition of the mfl paradox and its condition under expansion are examined in detail it is well known that the fundamental understanding of problems on mfl paradox is of paramount importance to applications of resource allotments and optimal resource management and correspondingly that the information science and technology advancement play a rule to resource allotments and resource option in management problems in fact they are dependent and interwinded

a matrix approach to latticized linear programming with fuzzy relation inequality constraints this paper investigates the latticized linear programming that is subject to the fuzzy relation inequality fri constraints with the max min composition by using the semi tensor product method and proposes a matrix approach to this problem first the resolution of the fri is studied and it is proved that all the minimal solutions and the unique maximum solution are within the finite parameter set solutions based on this and using the semi tensor product solving fris is converted to solving a set of algebraic inequalities and some new results on the resolution of fris are presented second the latticized linear programming that is subject to the fri constraints is solved by taking the following two key steps 1 the optimal value is obtained by calculating the minimum value of the objective function among all the minimal solutions to the fri constraints and 2 the optimal solution set is obtained by solving the fuzzy relation equation that is generated by letting the objective function equal to the optimal value the study of illustrative examples shows that the new results that are obtained in this paper are very effective in solving the latticized linear programming subject to the fri constraints

a new algorithm for minimizing a linear objective function with fuzzy relation equation constraints in this paper we study the problem of minimizing a linear objective function subject to a fuzzy system constraint by utilizing the fuzzy system s compact system we define k form chained solutions and spaces of k form chained solutions then we prove that any minimal solution of the fuzzy system is a k form chained solution by using an operation method expressed in tables we sieve out basic solutions from the spaces of k form chained solutions finally we obtain optimal solutions of the studied problem from the obtained basic solutions examples are provided to show that our algorithm is simple and convenient

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

an extension of some theorems of l barreira and c valls for the nonuniform exponential dichotomous evolution operators abstract the aim of this paper is to study the connection between the non uniform exponential dichotomy of a non uniform exponentially bounded strongly continuous evolution family and the admissibility of some function spaces thus there are extended recent results established by l barreira and c valls 2010 2011 in 10 and 11

promotion of cooperation by payoff noise in a 2 2 game a series of numerical simulations of a 2 ifmmode times else texttimes fi 2 symmetric game on a network examined whether payoff matrix noise promotes cooperation as reported initially by perc new j phys 8 22 2006 agents have no memory they offer cooperation c or defection d we assume that the network is time invariable the effect of payoff matrix noise pmn is measured by a simulated payoff difference between a normal network game and a network game with pmn the effect of pmn appears only when a local strategy adaptation is implemented for example a network game with imitation dynamics the influence of pmn becomes more significant with a larger stochastic deviation and less significant in a larger degree network one reason for pmn s effectiveness is the local strategy adaptation mechanism which helps both the preservation and fixation of c agents and not that the payoff matrix noise makes a dilemma game into a trivial dilemma free game

fuzzy linear systems abstract fuzzy linear systems of equations and inequalities over a bounded chain are studied both from theoretical and computational points of view a unified approach is presented for solving such systems completed with polynomial time algorithms the main results are concerned with establishing the consistency of the system computing all kinds of solutions or marking the contradictory equations respectively inequalities if the system is inconsistent applications in fuzzy linear programming multivalued logic fuzzy matrix equations or inequalities fuzzy relation equations or inclusions fuzzy automata and medicine are presented

cooperation in two dimensional mixed games evolutionary game theory is a common framework to study the evolution of cooperation where it is usually assumed that the same game is played in all interactions here we investigate a model where the game that is played by two individuals is uniformly drawn from a sample of two different games using the master equation approach we show that the random mixture of two games is equivalent to play the average game when i the strategies are statistically independent of the game distribution and ii the transition rates are linear functions of the payoffs we also use monte carlo simulations in a two dimensional lattice and mean field techniques to investigate the scenario when the two above conditions do not hold we find that even outside of such conditions several quantities characterizing the mixed games are still the same as the ones obtained in the average game when the two games are not very different

minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations this paper considers the following optimization problem minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations using max product composition we first show some properties of minimal solutions of a system of fuzzy relation equations with max product composition and present some rules for reducing the original problem then we derive an algorithm which enables us to find all the optimal solutions for solving the optimization problem and provide some numerical examples to illustrate

linear optimization problem constrained by fuzzy max min relation equations fang and li introduced the optimization model with a linear objective function and constrained by fuzzy max min relation equations they converted this problem into a 0 1 integer programming problem and solved it using the jump tracking branch and bound method subsequently wu et al improved this method by providing an upper bound on the optimal objective value and presented three rules for simplifying the computation of an optimal solution this work presents new theoretical results concerning this optimization problem they include an improved upper bound on the optimal objective value improved rules for simplifying the problem and a rule for reducing the solution tree accordingly an accelerated approach for finding the optimal objective value is presented and represents an improvement on earlier approaches its potential applications are discussed

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

resolution of fuzzy relational equations method algorithm and software with applications analytical methods and algorithms for inverse problem resolution of fuzzy linear systems of equations in some bl algebras godel algebra in case of max min and min max compositions and goguen algebra in case of max product composition are presented algorithms with software realization for solving fuzzy linear systems of equations are proposed applications include fuzzy optimization with fuzzy linear systems of equation constraint fuzzy machines and covering problem

ordering in spatial evolutionary games for pairwise collective strategy updates evolutionary 2 ifmmode times else texttimes fi 2 games are studied with players located on a square lattice during the evolution the randomly chosen neighboring players try to maximize their collective income by adopting a random strategy pair with a probability dependent on the difference of their summed payoffs between the final and initial states assuming quenched strategies in their neighborhood in the case of the anticoordination game this system behaves like an antiferromagnetic kinetic ising model within a wide region of social dilemmas this dynamical rule supports the formation of similar spatial arrangement of the cooperators and defectors ensuring the optimum total payoff if the temptation to choose defection exceeds a threshold value dependent on the sucker s payoff the comparison of the results with those achieved for pairwise imitation and myopic strategy updates has indicated the relevant advantage of pairwise collective strategy update in the maintenance of cooperation

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

role separating ordering in social dilemmas controlled by topological frustration three is a crowd is an old proverb that applies as much to social interactions as it does to frustrated configurations in statistical physics models accordingly social relations within a triangle deserve special attention with this motivation we explore the impact of topological frustration on the evolutionary dynamics of the snowdrift game on a triangular lattice this topology provides an irreconcilable frustration which prevents anticoordination of competing strategies that would be needed for an optimal outcome of the game by using different strategy updating protocols we observe complex spatial patterns in dependence on payoff values that are reminiscent to a honeycomb like organization which helps to minimize the negative consequence of the topological frustration we relate the emergence of these patterns to the microscopic dynamics of the evolutionary process both by means of mean field approximations and monte carlo simulations for comparison we also consider the same evolutionary dynamics on the square lattice where of course the topological frustration is absent however with the deletion of diagonal links of the triangular lattice we can gradually bridge the gap to the square lattice interestingly in this case the level of cooperation in the system is a direct indicator of the level of topological frustration thus providing a method to determine frustration levels in an arbitrary interaction network

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

segregation process and phase transition in cyclic predator prey models with an even number of species we study a spatial cyclic predator prey model with an even number of species for n 4 6 and 8 that allows the formation of two defensive alliances consisting of the even and odd label species the species are distributed on the sites of a square lattice the evolution of spatial distribution is governed by iteration of two elementary processes on neighboring sites chosen randomly if the sites are occupied by a predator prey pair then the predator invades the prey s site otherwise the species exchange their sites with a probability x for low x values a self organizing pattern is maintained by cyclic invasions if x exceeds a threshold value then two types of domain grow up that are formed by the odd and even label species respectively monte carlo simulations indicate the blocking of this segregation process within a range of x for n 8

ising model versus normal form game abstract the 2 spin ising model in statistical mechanics and the 2 2 normal form game in game theory are compared all configurations allowed by the second are recovered by the first when the only concern is about nash equilibria but it holds no longer when pareto optimum considerations are introduced as in the prisoner s dilemma this gap can nevertheless be filled by adding a new coupling term to the ising model even if that term has up to now no physical meaning an individual complete bilinear objective function is thus found to be sufficient to reproduce all possible configurations of a 2 2 game using this one to one mapping new perspectives for future research in both fields can be envisioned

solution algorithms for fuzzy relational equations with max product composition the conditions for the existence of an inverse solution to the max min composition of fuzzy relational equations have been well documented since the original work by sanchez 30 31 these same existence theorems have been extended to the t norm composition of relational equations in which the max product composition is a member 5 13 26 several studies 8 15 24 33 34 38 have shown that the max min operator may not always be the most desirable fuzzy relational composition and in fact the max product operator was superior in these instances this paper reviews the algorithms necessary to determine the complete solution of the inverse for fuzzy relational equations with max product composition

an algorithm for solving optimization problems with fuzzy relational inequality constraints an algorithm for solving a kind of optimization problems with fuzzy relational inequalities was proposed by guo and xia 6 however it is too expensive to verify the optimal condition in this paper some rules for reducing these problems are proposed and the relationship between minimal solutions and fri paths is also given these lead to a new algorithm for solving this kind of problems numerical experiments are presented for illustrating the efficiency of the proposed algorithm

minimizing a linear objective function under a fuzzy max t norm relation equation constraint the work examines the feasibility of minimizing a linear objective function subject to a max t fuzzy relation equation constraint where t is a continuous archimedean t norm conventional methods for solving this problem are significantly improved by first separating the problem into two sub problems according to the availability of positive coefficients this decomposition is thus more easily handled than in previous literature next based on use of the maximum solution of the constraint equation the sub problem with non positive coefficients is solved and the size of the sub problem with positive coefficients reduced as well this step is unique among conventional methods owing to its ability to determine as many optimal variables as possible additionally several rules are developed for simplifying the remaining problem finally those undecided optimal variables are obtained using the covering problem rather than the branch and bound methods three illustrative examples demonstrate that the proposed approach outperforms conventional schemes its potential applications are discussed as well

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

lower solutions of systems of fuzzy equations lower solutions of a system of fuzzy equations are determined by lower solutions of each equation the result is applied to fuzzy relation equations so that they are decomposed to simple fuzzy relation equations

exponential dichotomy of evolution equations and admissibility of function spaces on a half line consider an evolution family u u t s ts0 on a half line r and an integral equation u t u t s u s t s u t f d we characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of lp type the lorentz spaces lp q and many other function spaces occurring in interpolation theory we then apply our results to study the robustness of the exponential dichotomy of evolution families on a half line under small perturbations

phase diagrams for an evolutionary prisoner s dilemma game on two dimensional lattices the effects of payoffs and noise on the maintenance of cooperative behavior are studied in an evolutionary prisoner s dilemma game with players located on the sites of different two dimensional lattices this system exhibits a phase transition from a mixed state of cooperators and defectors to a homogeneous one where only the defectors remain alive using monte carlo simulations and the generalized mean field approximations we have determined the phase boundaries critical points separating the two phases on the plane of the temperature noise and temptation to choose defection in the zero temperature limit the cooperation can be sustained only for those connectivity structures where three site clique percolation occurs

cooperation in the noisy case prisoner s dilemma game on two types of regular random graphs we have studied an evolutionary prisoner s dilemma game with players located on two types of random regular graphs with a degree of 4 the analysis is focused on the effects of payoffs and noise temperature on the maintenance of cooperation when varying the noise level and or the highest payoff the system exhibits a second order phase transition from a mixed state of cooperators and defectors to an absorbing state where only defectors remain alive for the random regular graph and bethe lattice the behavior of the system is similar to those found previously on the square lattice with nearest neighbor interactions although the measure of cooperation is enhanced by the absence of loops in the connectivity structure for low noise the optimal connectivity structure is built up from randomly connected triangles

optimization of fuzzy relation equations with max product composition abstract an optimization problem with a linear objective function subject to a system of fuzzy relation equations using max product composition is considered since the feasible domain is non convex traditional linear programming methods cannot be applied we study this problem and capture some special characteristics of its feasible domain and the optimal solutions some procedures for reducing the original problem are presented the problem is transformed into a 0 1 integer program which is then solved by the branch and bound method for illustration purpose an example of the procedures is provided

latticized linear optimization on the unit interval this paper considers the latticized linear optimization llo problem and its variants which are a special class of optimization problems constrained by fuzzy relational equations or inequalities we show that an optimal solution to such a problem can be obtained in polynomial time as long as the objective function is a max separable function with continuous monotone components we further show that the set of all optimal solutions is fully determined by one maximum optimal solution and a finite number of minimal optimal solutions the maximum optimal solution can be constructed in polynomial time once the optimal objective value is known while the detection of all minimal optimal solutions in an efficient manner remains as a challenging problem the relation between llo and max separable optimization and related issues are also investigated

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

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

eigenspace structure of a max prod fuzzy matrix the eigenvectors of a fuzzy matrix correspond to steady states of a complex discrete events system characterized by the given transition matrix and fuzzy state vectors the descriptions of the eigenspace the set of all eigenvectors for matrices in the max min max ukasiewicz or max drast fuzzy algebra have been presented in previous papers the eigenspace of a fuzzy matrix in the max prod algebra is investigated in this paper first necessary and sufficient conditions are shown under which the eigenspace restricted to increasing eigenvectors of a given matrix is non empty and the structure of the increasing eigenspace is described then using simultaneous row and column permutations of the matrix the complete characterization of the whole eigenspace structure of a given fuzzy matrix is shown the details for matrices of order 3 are only presented the method works analogously for square matrices of higher orders with rapidly increasing complexity of the formulas

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

inertial manifolds for semi linear parabolic equations in admissible spaces we prove the existence of inertial manifolds for the solutions to the semi linear parabolic equation du t dt au t f t u when the partial differential operator a is positive definite and self adjoint with a discrete spectrum having a sufficiently large distance between some two successive points of the spectrum and the nonlinear forcing term f satisfies the lipschitz conditions on the domain d a 0 1 i e f t x f t y t a x y and f t x t 1 a x where t belongs to one of admissible function spaces containing wide classes of function spaces like lp spaces the lorentz spaces lp q and many other function spaces occurring in interpolation theory

resolution of finite fuzzy relation equations abstract general schemes for solving fuzzy relation equations with finite sets are derived in this paper the relationship between the results derived here and those of previous publications regarding this subject is also discussed

discretized characterizations of exponential dichotomy of linear skew product semiflows over semiflows using the method of discretization we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear skew product semiflows we then apply the obtained results to consider the roughness of exponential dichotomy of linear skew product semiflows

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

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

stable manifolds for semi linear evolution equations and admissibility of function spaces on a half line abstract consider an evolution family u u t s t s 0 on a half line r and a semi linear integral equation u t u t s u s s t u t f u d we prove the existence of stable manifolds of solutions to this equation in the case that u t s t s 0 has an exponential dichotomy and the nonlinear forcing term f t x satisfies the non uniform lipschitz conditions f t x 1 f t x 2 t x 1 x 2 for being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of l p type the lorentz spaces l p q and many other function spaces occurring in interpolation theory

evolutionary mixed games in structured populations cooperation and the benefits of heterogeneity evolutionary games on networks traditionally involve the same game at each interaction here we depart from this assumption by considering mixed games where the game played at each interaction is drawn uniformly at random from a set of two different games while in well mixed populations the random mixture of the two games is always equivalent to the average single game in structured populations this is not always the case we show that the outcome is in fact strongly dependent on the distance of separation of the two games in the parameter space effectively this distance introduces payoff heterogeneity and the average game is returned only if the heterogeneity is small for higher levels of heterogeneity the distance to the average game grows which often involves the promotion of cooperation the presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin and they also emphasize the importance of the population structure in amplifying facilitators of cooperation

solving nonlinear optimization problems with fuzzy relation equation constraints abstract an optimization model with a nonlinear objective function subject to a system of fuzzy relation equations is presented since the solution set of the fuzzy relation equations is in general a non convex set when it is not empty conventional nonlinear programming methods are not ideal for solving such a problem in this paper a genetic algorithm ga is proposed this ga is designed to be domain specific by taking advantage of the structure of the solution set of fuzzy relation equations the individuals from the initial population are chosen from the feasible solution set and are kept within the feasible region during the mutation and crossover operations the construction of test problems is also developed to evaluate the performance of the proposed algorithm

an efficient solution procedure for fuzzy relation equations with max product composition we study a system of fuzzy relation equations with max product composition and present an efficient solution procedure to characterize the whole solution set by finding the maximum solution as well as the complete set of minimal solutions instead of solving the problem combinatorially the procedure identifies the nonminimal solutions and eliminates them from the set of minimal solutions

integral manifolds for partial functional differential equations in admissible spaces on a half line abstract in this paper we investigate the existence of stable and center stable manifolds for solutions to partial functional differential equations of the form u t a t u t f t u t t 0 when its linear part the family of operators a t t 0 generates the evolution family u t s t s 0 having an exponential dichotomy or trichotomy on the half line and the nonlinear forcing term f satisfies the lipschitz condition i e f t u t f t v t t u t v t c where u t v t c c r 0 x and t belongs to some admissible function space on the half line our main methods invoke lyapunov perron methods and the use of admissible function spaces

group formation in the spatial public goods game with continuous strategies abstract we study group formation in the spatial public goods game with continuous strategies an active group will become inactive with high low probability if the collective contribution is below above a threshold value meanwhile an inactive group recovers to be active with some probability we have found that the cooperation level and the average payoff of players are maximized at moderate values of the threshold and the recovery rate spatial distributions of strategies and active groups are plotted to understand the evolution of cooperation