hierarchical construction of bounded solutions in critical regularity spaces we construct uniformly bounded solutions for the equations divu d f and curlu d f in the critical cases f 2 l d t d r and f2 l 3 t 3 r 3 respectively criticality in this context manifests itself by the lack of a linear solution operator mapping l d 7 l 1 t d thus the intriguing aspect here is that although the problems are linear construction of their solutions is not our constructions are special cases of a general framework for solving linear equations of the form lu d f where l is a linear operator densely defined in banach space b with a closed range in a proper subspace of lebesgue space l p and with an injective dual l the solutions are realized in terms of a multiscale hierarchical representation u d p 1 jd1 uj interesting for its own sake here the uj s are constructed recursively as minimizers of ujc1d arg min u kukbc jc1krj luk p lp

nucleation of instability of the meissner state of 3 dimensional superconductors this paper concerns a nonlinear partial differential system in a 3 dimensional domain involving the operator curl2 which is a simplified model used to examine nucleation of instability of the meissner state of a superconductor as the applied magnetic field reaches the superheating field we derive a priori c 2 estimates for a weak solution h the curl of the magnetic potential and determine the location of the maximal points of curlh which correspond to the nucleation of instability of the meissner state we show that if the penetration length is small the solution exhibits a boundary layer if the applied magnetic field is homogeneous curlh is maximal around the points on the boundary where the applied field is tangential to the surface

the form boundedness criterion for the laplacian operator abstract in this paper we characterize the class of measurable functions or more generally real or complex valued distributions f such that the commutator operator c f f acting from h 1 r d to h 1 r d in other words we give a complete solution to the problem for the boundedness of the laplacian which is fundamental to quantum mechanics

renewal reward processes with fuzzy rewards and their applications to t age replacement policies the application of fuzzy set theory to renewal reward processes is proposed in this paper the reward is modeled as a fuzzy random variable a theorem which presents the long run average fuzzy reward per unit time is stated a procedure to obtain the best t age replacement policy with fuzzy cost structure is developed the original problem is transformed into a nonlinear program in order to evaluate the membership of the long run average fuzzy cost per unit time

solvability of the divergence equation implies john via poincare inequality abstract let r 2 be a bounded simply connected domain we show that for a fixed every p 1 the divergence equation div v f is solvable in w 0 1 p 2 for every f l 0 p if and only if is a john domain if and only if the weighted poincare inequality u x u q d x c u x q dist x q d x holds for some every q 1 this gives a positive answer to a question raised by russ 2013 in the case of bounded simply connected domains in higher dimensions similar results are proved under some additional assumptions on the domain in question

some properties of fuzzy random renewal processes fuzzy random variable is a measure function from a probability space to a collection of fuzzy variables based on the fuzzy random theory this paper addresses some properties of fuzzy random renewal processes generated by a sequence of independent and identically distributed iid fuzzy random interarrival times the relationship between the expected value of the fuzzy random renewal variable and the distribution functions of the alpha pessimistic values and alpha optimistic values of the interarrival times is discussed furthermore the fuzzy random style of renewal equation is provided finally fuzzy random blackwell s renewal theorem and smith s key renewal theorem are also given

solutions of the divergence operator on john domains if rn is a bounded domain the existence of solutions u w01 p of divu f for f lp with vanishing mean value and 1 p is a basic result in the analysis of the stokes equations it is known that the result holds when is a lipschitz domain and that it is not valid for domains with external cusps in this paper we prove that the result holds for john domains our proof is constructive the solution u is given by an explicit integral operator acting on f to prove that u w01 p we make use of the calderon zygmund singular integral operator theory and the hardy littlewood maximal function for domains satisfying the separation property introduced in s buckley p koskela sobolev poincare implies john math res lett 2 5 1995 577 593 and 1 p n we also prove a converse result thus characterizing in this case the domains for which a continuous right inverse of the divergence exists in particular our result applies to simply connected planar domains because they satisfy the separation property

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

memory based snowdrift game on networks we present a memory based snowdrift game mbsg taking place on networks we found that when a lattice is taken to be the underlying structure the transition of spatial patterns at some critical values of the payoff parameter is observable for both four and eight neighbor lattices the transition points as well as the styles of spatial patterns can be explained by local stability analysis in sharp contrast to previously reported results cooperation is promoted by the spatial structure in the mbsg interestingly we found that the frequency of cooperation of the mbsg on a scale free network peaks at a specific value of the payoff parameter this phenomenon indicates that properly encouraging selfish behaviors can optimally enhance the cooperation the memory effects of individuals are discussed in detail and some nonmonotonous phenomena are observed on both lattices and scale free networks our work may shed some new light on the study of evolutionary games over networks

on inequalities of korn friedrichs and babuska aziz korn s inequalities for integrals of quadratic functionals subject to certain side conditions have played a fundamental role in the development of elasticity theory see e g i 14 the korn inequalities are essential in establishing coerciveness of the differential operators of linear elastostatics and thus form the basis for existence results in that theory 9 among other areas in elasticity where consideration of korn s inequalities arise we cite for example fundamental studies on the mathematical foundations of finite elements 15 17 stability theory 11 and qualitative analyses of solution behavior such as those involved in the investigation of saint venant s principle 18 20 in the latter two areas of application in particular information on the optimal constants appearing in the inequalities the korn constants is of importance thus several investigations have been concerned with evaluating the korn constants for specific domains and with obtaining bounds for these constants 4 6 8 12 13 our purpose in this paper is to draw attention to an equivalence for two dimensional simply connected domains between korn s inequality in the second case see w 2 here and two other inequalities for quadratic functionals which have received comparatively little attention in the literature the first of these is an inequality due apparently to k o frmorichs 2 between the real and imaginary parts of an analytic function or equivalently between a harmonic function and its conjugate defined on a two dimensional domain r thus if w z f ig z xl ix2 is analytic on r friedrichs inequality reads

expected value of fuzzy variable and fuzzy expected value models this paper will present a novel concept of expected values of fuzzy variables which is essentially a type of choquet integral and coincides with that of random variables in order to calculate the expected value of general fuzzy variable a fuzzy simulation technique is also designed finally we construct a spectrum of fuzzy expected value models and integrate fuzzy simulation neural network and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models

bourgain brezis inequalities on symmetric spaces of non compact type abstract let m be a global riemannian symmetric space of non compact type we prove a duality estimate for pairings of divergence free l 1 vector fields with vector fields in a critical sobolev space on m m f d v c f l 1 d v l m d v this estimate provides a remedy for the failure of a critical sobolev embedding on such symmetric spaces

form boundedness of the general second order differential operator we give explicit necessary and sufficient conditions for the boundedness of the general second order differential operator i j ln aij i j j 1nbj j c with real or complex valued distributional coefficients aij bj and c acting from the sobolev space w1 2 n to its dual w 1 2 n this enables us to obtain analytic criteria for the fundamental notions of relative form boundedness compactness and infinitesimal form boundedness of c with respect to the laplacian on l 2 n in particular we establish a complete characterization of the form boundedness of the schrodinger operator i a 2 q with magnetic vector potential a lloc2 n n and q d n 2006 wiley periodicals inc

korn s inequalities and their applications in continuum mechanics korn s inequalities have played a central role in the development of linear elasticity not only in connection with the basic theoretical issues such as existence and uniqueness but also in a variety of applications the korn inequalities and other related inequalities for integrals of quadratic functionals also arise in the analysis of viscous incompressible fluid flow the dimensionless optimal constants appearing in these inequalities the korn constants depend only on the shape of the domains of concern information on the geometric dependence of these constants is essential in applications in this review article we summarize the major results on korn s inequalities for bounded domains in two and three dimensions with emphasis on results concerning the korn constants some applications in continuum mechanics are also described

gauss green theorem for weakly differentiable vector fields sets of finite perimeter and balance laws we analyze a class of weakly differentiable vector fields f w r n r n with the property that f 2 l 1 and div f is a signed radon measure these fields are called bounded divergence measure fields the primary focus of our inves tigation is to introduce a suitable notion of the normal trace of any divergence measure field f over the boundary of an arbitrary set of finite perimeter that en sures the validity of the gauss green theorem to achieve this we first establish a fundamental approximation theorem which states that given a signed radon measurethat is absolutely continuous with respect to h n 1 on r n a ny set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure theoretic interior of the set with respect to the mea sure kk the total variation measure we employ this approximation theorem to derive the normal trace of f on the boundary of any set of finite perimeter e as the limit of the normal traces of f on the boundaries of the approximate sets with smooth boundary so that the gauss green theorem for f holds one with these results we analyze the cauchy flux that is bounded by a nonnegative radon measure over any oriented surface i e an n 1 dimensional surface that is a part of the boundary of a set of finite perimeter and thereby develop a general mathematical formulation of the physical principle of the balance law through the cauchy flux finally we apply this framework to the derivation of systems of balance laws with measure valued source terms from the formulation of the balance law this framework also allows the recovery of cauchy entropy flux through the lax entropy inequality for entropy solutions of hyperbolic con servation laws c 2008 wiley periodicals inc

a note on quasihyperbolic boundary conditions and poincare domains we establish optimal poincare inequalities under a logarithmic growth condition on the quasihyperbolic metric

enhanced cooperation and harmonious population in an evolutionary n person snowdrift game abstract the n person evolutionary snowdrift game nesg is generalized to study the effects of the additional benefit to all agents in a competing group of size n resulting from an earlier completion of a task when more agents are willing to share the work following replicator dynamics an equation that can be used to solve for the steady state frequency of cooperation x in a well mixed population as a function of the parameters representing the cost to benefit ratio c b additional reward w b and n is derived cooperation is enhanced in general for w 0 and a stable state with all cooperative agents allc state emerges for small groups n and small c b in contrast such a harmonious allc state does not exist in the original nesg for c 0 the condition for the existence of an allc state is estimated to be n 1 c w and good agreement is found when compared with numerical solutions the observed disappearance of an allc state and the drop in x with n are discussed in terms of the change in the stability of the allc state as n increases previous results in nesg are recovered as the w 0 case of our model thus the pleasure of enjoying the completed task earlier promotes cooperation and the effects are more pronounced when small group collective interactions are considered

using fuzzy random variables in life annuities pricing this paper develops life annuity pricing with stochastic representation of mortality and fuzzy quantification of interest rates we show that modelling the present value of annuities with fuzzy random variables allows quantifying their expected price and risk resulting from the uncertainty sources considered so we firstly describe fuzzy random variables and define some associated measures the mathematical expectation the variance distribution function and quantiles secondly we show several ways to estimate the discount rates to price annuities subsequently the present value of life annuities is modelled with fuzzy random variables we finally show how an actuary can quantify the price and the risk of a portfolio of annuities when their present value is given by means of fuzzy random variables

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

a new approach to counterexamples to l 1 estimates korn s inequality geometric rigidity and regularity for gradients of separately convex functions the derivation of counterexamples to l1 estimates can be reduced to a geometric decomposition procedure along rank one lines in matrix space we illustrate this concept in two concrete applications firstly we recover a celebrated and rather complex counterexample by ornstein proving the failure of korn s inequality and of the corresponding geometrically nonlinear rigidity result in l1 secondly we construct a function f 2 which is separately convex but whose gradient is not in bvloc in the sense that the mixed derivative 2f x1 x2 is not a bounded measure

a theorem of renewal process for fuzzy random variables and its application the fuzzy set was introduced by zadeh 1965 and the concept of fuzzy random variables was provided by kwakernaak 1981 sequences of independent and identical distributed fuzzy random variables were considered by kruse 1982 he also showed the strong law of large numbers for fuzzy random variables in this paper we consider the stochastic process for fuzzy random variables and prove a theorem for fuzzy rate of a fuzzy renewal process

a limiting case for the divergence equation we consider the equation text div mathbb y f with f a zero average function on the torus mathbb t d in their seminal paper bourgain and brezis j am math soc 16 2 393 426 2003 electronic proved the existence of a solution mathbb y in w 1 d cap l infty for a datum f in l d we extend their result to the critical sobolev spaces w s p with s 1 p d and p ge 2 more generally we prove a similar result in the scale of triebel lizorkin spaces we also consider the equation text div mathbb y f in a bounded domain varomega subject to zero dirichlet boundary condition

three strategy n person snowdrift game incorporating loners the nn person snowdrift game is generalized to incorporate a third strategy in addition to the cooperative cc and non cooperative dd strategies a strategy ll representing a loner behavior is introduced agents taking on the ll strategy ll agents do not contribute to the game as the cc agents do but they do not take advantage of the cc agents instead they would rather settle with a fixed payoff ll dynamical equations governing the time evolution of the frequencies of the strategies in a well mixed population are derived the dynamics and the frequencies of the steady state reveal the rich behavior resulting from the interplay between the payoff rr which promotes the non cooperative behavior and ll detailed studies on how a system evolves indicated that the steady state could be an alll allc or c d state depending on the parameters rr ll and group size nn in contrast only a c d state results for r 0r 0 and an allc state is possible only at r 0r 0 without the strategy ll with the strategy ll the allc phase occupies a finite though tiny region of the rr ll parameter space the ll agents play an important role in the dynamics leading to the allc phase they help eliminate the dd strategy in the transient and later only to be replaced by the cc strategy phase diagrams in the rr ll space are presented for different values of nn the strategy ll plays two roles it leads to an alll phase and helps give an allc phase an algorithm for simulating the model numerically is described and validated the algorithm will be useful in studying our model in various structured populations

quasihyperbolic boundary conditions and poincare domains we prove that a domain in bbb r n whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient beta le 1 is a q p poincare domain for all p and q satisfying p in 1 infty cap n n beta n and q in p beta p where p np n p denotes the sobolev conjugate exponent an elementary example shows that the given ranges for p and q are sharp the proof makes use of estimates for a variational capacity when p 2 we give an application to the solvability of the neumann problem on domains with irregular boundaries we also discuss the relationship between this growth condition on the quasihyperbolic metric and the s john condition

modelling redundancy allocation for a fuzzy random parallel series system due to subjective judgment imprecise human knowledge and perception in capturing statistical data the real data of lifetimes in many systems are both random and fuzzy in nature based on the fuzzy random variables that are used to characterize the lifetimes this paper studies the redundancy allocation problems to a fuzzy random parallel series system two fuzzy random redundancy allocation models fr ram are developed through reliability maximization and cost minimization respectively some properties of the fr ram are obtained in which an analytical formula of reliability with convex lifetimes is derived and the sensitivity of the reliability is discussed to solve the fr rams we first address the computation of reliability a random simulation method based on the derived analytical formula is proposed to compute the reliability with convex lifetimes as for the reliability with nonconvex lifetimes the technique of fuzzy random simulation together with the discretization method of fuzzy random variable is employed to compute the reliability and a convergence theorem of the fuzzy random simulation is proved subsequently we integrate the computation approaches of the reliability and genetic algorithm ga to search for the approximately optimal redundancy allocation of the models finally some numerical examples are provided to illustrate the feasibility of the solution algorithm and quantify its effectiveness

fuzzy random chance constrained programming by fuzzy random programming we mean the optimization theory dealing with fuzzy random decision problems this paper presents a new concept of chance of fuzzy random events and constructs a general framework of fuzzy random chance constrained programming we also design a spectrum of fuzzy random simulations for computing uncertain functions arising in the area of fuzzy random programming to speed up the process of handling uncertain functions we train a neural network to approximate uncertain functions based on the training data generated by fuzzy random simulation finally we integrate the fuzzy random simulation neural network and genetic algorithm to produce a more powerful and effective hybrid intelligent algorithm for solving fuzzy random programming models and illustrate its effectiveness by some numerical examples

risk assessment of serious crime with fuzzy random theory this paper presents a novel approach for assessing the potential risk of serious crime events e g terrorist attack the modelling and assessment of such risk is carried out under uncertain circumstances because of both the randomness and fuzziness inherent in crime data the approach is based on fuzzy random theory that complements probability theory with an additional dimension of imprecision this allows for potential loss caused by a crime to be expressed as a fuzzy random variable crime risk is therefore estimated as the mean chance of a fuzzy random event where the resulting loss reaches a given confidence level the concept of the average loss per unit of time is also introduced in order to calculate the rate at which the loss may increase due to possible crime events the work is compared with typical existing approaches and supported with examples throughout that illustrate its utility

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

quantum strategies win in a defector dominated population a b s t r a c t quantum strategies are introduced into evolutionary games the agents using quantum strategies are regarded as invaders whose fraction generally is 1 of a population in contrast to the 50 of the population that are defectors in this paper the evolution of strategies on networks is investigated in a defector dominated population when three networks square lattice newman watts small world network and scale free network are constructed and three games prisoners dilemma snowdrift and stag hunt are employed as far as these three games are concerned the results show that quantum strategies can always invade the population successfully comparing the three networks we find that the square lattice is most easily invaded by agents that adopt quantum strategies however a scale free network can be invaded by agents adopting quantum strategies only if a hub is occupied by an agent with a quantum strategy or if the fraction of agents with quantum strategies in the population is significant

effects of dynamical grouping on cooperation in n person evolutionary snowdrift game a population typically consists of agents that continually distribute themselves into different groups at different times this dynamic grouping has recently been shown to be essential in explaining many features observed in human activities including social economic and military activities we study the effects of dynamic grouping on the level of cooperation in a modified evolutionary n person snowdrift game due to the formation of dynamical groups the competition takes place in groups of different sizes at different times and players of different strategies are mixed by the grouping dynamics it is found that the level of cooperation is greatly enhanced by the dynamic grouping of agents when compared with a static population of the same size as a parameter ensuremath beta which characterizes the relative importance of the reward and cost increases the fraction of cooperative players f c increases and it is possible to achieve a fully cooperative state analytically we present a dynamical equation that incorporates the effects of the competing game and group size distribution the distribution of cooperators in different groups is assumed to be a binomial distribution which is confirmed by simulations results from the analytic equation are in good agreement with numerical results from simulations we also present detailed simulation results of f c over the parameter space spanned by the probabilities of group coalescence ensuremath nu m and group fragmentation ensuremath nu p in the grouping dynamics a high ensuremath nu m and low ensuremath nu p promotes cooperation and a favorable reward characterized by a high ensuremath beta would lead to a fully cooperative state

on finite time ruin probabilities in a generalized dual risk model with dependence abstract in this paper we study the finite time ruin probability in a reasonably generalized dual risk model where we assume any non negative non decreasing cumulative operational cost function and arbitrary capital gains arrival process establishing an enlightening link between this dual risk model and its corresponding insurance risk model explicit expressions for the finite time survival probability in the dual risk model are obtained under various general assumptions for the distribution of the capital gains in order to make the model more realistic and general different dependence structures among capital gains and inter arrival times and between both are also introduced and corresponding ruin probability expressions are also given the concept of alarm time as introduced in das and kratz 2012 is applied to the dual risk model within the context of risk capital allocation extensive numerical illustrations are provided

a constructive proof of the fefferman stein decomposition of bmo rn let s vec r left left f r 1 f cdots r n f right f in h 1 left text r n text r right right subset h 1 left text r n text r n 1 right 22 1 begin gathered s vec r left vec g left g j right j 0 cdots n right in text bmo left text r n text r n 1 right sum limits j 0 n vec r j g j 0 text in bmo left text r n text r n 1 right left text r right end gathered 22 2 j 0

korn inequality on irregular domains abstract in this paper we study the weighted korn inequality on some irregular domains e g s john domains and domains satisfying quasihyperbolic boundary conditions examples regarding sharpness of the korn inequality on these domains are presented moreover we show that korn inequalities imply certain poincare inequality

a conformal korn inequality on holder domains abstract in this paper we derive a sharp conformal korn inequality on holder domains we also provide an example for the sharpness of the result this shows that the conformal korn inequality has a different behavior than the classical one on holder domains

cooperation in n person evolutionary snowdrift game in scale free barabasi albert networks abstract cooperation in the n person evolutionary snowdrift game nesg is studied in scale free barabasi albert ba networks due to the inhomogeneity of the network two versions of nesg are proposed and studied in a model where the size of the competing group varies from agent to agent the fraction of cooperators drops as a function of the payoff parameter the networking effect is studied via the fraction of cooperative agents for nodes with a particular degree for small payoff parameters it is found that the small k agents are dominantly cooperators while large k agents are of non cooperators studying the spatial correlation reveals that cooperative agents will avoid to be nearest neighbors and the correlation disappears beyond the next nearest neighbors the behavior can be explained in terms of the networking effect and payoffs in another model with a fixed size of competing groups the fraction of cooperators could show a non monotonic behavior in the regime of small payoff parameters this non trivial behavior is found to be a combined effect of the many agents with the smallest degree in the ba network and the increasing fraction of cooperators among these agents with the payoff for small payoffs

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

sharp a priori estimates for div curl systems in heisenberg groups abstract in this paper we prove a family of inequalities for differential forms in heisenberg groups h 1 and h 2 that are the natural counterpart of a class of div curl inequalities in de rham s complex proved by lanzani stein and bourgain brezis

weighted hardy inequalities abstract for bounded lipschitz domains d in r n it is known that if 1 p then for all 0 0 where 0 p 1 0 there is a constant c with d u x p dist x d p d x c d u x p dist x d d x for all u c 0 d we show that if d is merely assumed to be a bounded domain in r n that satisfies a whitney cube counting condition with exponent and has plump complement then the same inequality holds with 0 now taken to be p n n p n p 2 n further we extend the known results see h brezis m marcus hardy s inequalities revisited dedicated to ennio de giorgi ann scuola norm sup pisa cl sci 4 25 1997 1998 217 237 m hoffmann ostenhof t hoffmann ostenhof a laptev a geometrical version of hardy s inequality j funct anal 189 2002 537 548 j tidblom a geometrical version of hardy s inequality for w 1 p proc amer math soc 132 2004 2265 2271 concerning the improved hardy inequality d u x p dist x d p d x d p n d u x p d x c d u x p d x c c n p by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains

evolution of cooperation in well mixed n person snowdrift games abstract we study the time evolution of cooperation in a recently proposed n person evolutionary snowdrift game by focusing on the details of the evolutionary dynamics it is found that the analytic solution for the equilibrium fraction of cooperators as given previously by the replicator dynamics stems from a balance between the terms the cost to contribute to a common task and the risk in refusing to participate in a common task analytic expressions for these two terms are given and their magnitudes are studied over the whole range of parameter space away from equilibrium it is the imbalance between these terms that drives the system to equilibrium a continuous time first order differential equation for the degree of cooperation is derived for arbitrary interacting group size n and cost to benefit ratio analytic solutions to the time evolution of cooperation for the cases of n 2 and n 3 are obtained with results in good agreement with those obtained by numerical simulations for arbitrary n numerical solutions to the equation give the time evolution of cooperation with the long time limit giving the equilibrium fraction of cooperators

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