integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

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

fuzzy random renewal reward process and its applications this paper studies a renewal reward process with fuzzy random interarrival times and rewards under the independence associated with any continuous archimedean t norm the interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are independent fuzzy variables under these conditions some limit theorems in mean chance measure are derived for fuzzy random renewal rewards in the sequel a fuzzy random renewal reward theorem is proved for the long run expected reward per unit time of the renewal reward process the renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory finally some application examples are provided to illustrate the utility of the result

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

tridiagonal matrices orthogonal polynomials and diophantine relations i it is well known that the eigenvalues of tridiagonal matrices can be identified with the zeros of polynomials satisfying three term recursion relations and being therefore members of an orthogonal set a class of such polynomials is identified some of which feature zeros given by simple formulae involving integer numbers in the process certain neat formulae are also obtained which perhaps deserve to be included in standard compilations since they involve classical polynomials such as the jacobi polynomials and other named polynomials

sufficient number of integrals for the pth order lyness equation in this communication we present a sufficient number of first integrals for the lyness equation of arbitrary order we first use the staircase method quispel et al 1991 physica a 173 243 66 to construct integrals of a derivative equation of the lyness equation closed form expressions for the integrals are given based on a non commutative vieta expansion the integrals of the lyness equation then follow directly from these integrals previously found integrals for the lyness equation arise as special cases of our new set of integrals

involutivity of integrals of sine gordon modified kdv and potential kdv maps closed form expressions in terms of multi sums of products have been given in tran et al 2009 j phys a math theor 42 225201 and van der kamp et al 2007 j phys a math theor 39 12789 98 of integrals of sine gordon modified korteweg de vries and potential korteweg de vries maps obtained as so called p 1 travelling wave reductions of the corresponding partial difference equations we prove the involutivity of these integrals with respect to recently found symplectic structures for those maps the proof is based on explicit formulae for the poisson brackets between multi sums of products

the toda molecule equation and the 3 algorithm one of the well known convergence acceleration methods the e algorithm is investigated from the viewpoint of the toda molecule equation it is shown that the error caused by the algorithm is evaluated by means of solutions for the equation the acceleration algorithm based on the discrete toda molecule equation is also presented

three point partial difference equations linearizable by local and nonlocal transformations we consider a class of nonlinear partial difference equations defined on three points of a plane lattice we construct conditions for this class of partial difference equations to be linearizable through a point or a cole hopf transformation using these conditions we are able to classify all multilinear linearizable equations belonging to this class

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

renewal process with t related fuzzy inter arrival times and fuzzy rewards in this paper we consider a renewal process in which the inter arrival times and rewards are characterized as fuzzy variables under t norm based fuzzy operations a t related fuzzy renewal theorem and a fuzzy renewal reward theorem are proved using a law of large numbers for fuzzy variables

initial value problems for quad equations we describe a method to construct well posed initial value problems ivps for not necessarily integrable equations on not necessarily simply connected quad graphs although the method does not always provide a well posed ivp not all quad graphs admit well posed ivps it is effective in the class of rhombic embeddable quad graphs

random fuzzy renewal process abstract this paper attempts to discuss a random fuzzy renewal process based on random fuzzy theory the interarrival times are characterized as nonnegative random fuzzy variables which is a more reasonable consideration in the real world under this setting the rate of the random fuzzy renewal process is discussed and a random fuzzy elementary renewal theorem is presented furthermore the expected value of renewals in an arbitrary interval is investigated and blackwell s theorem in random fuzzy sense is also established

the quotient difference algorithm and the pade table an alternative form and a general continued fraction the quotient difference algorithm is applied to a given power series in a modified way and various continued fractions provided by the algorithm are described in terms of their relationships with the pade table for the power series in particular a general continued fraction whose convergents form any chosen combination of horizontal or vertical connected sequences of pade approximants is introduced 1 introduction in 4 gragg gives a substantial review of some of the properties of the pade table for a given power series ek o k and the relations between pade approximants and other areas of numerical analysis the earliest and as gragg suggests perhaps the most significant of these related areas is the theory of continued fractions in particular it is the so called corresponding fractions for the power series that then provide the link between the pade table and the quotient dif ference algorithm of rutishauser one of the many applications of the quotient difference algorithm is to obtain the coefficients of the corresponding fractions from those of the power series and the convergents of these continued fractions form staircases in the pade table for the series in this paper the quotient difference algorithm is taken in a form that is slightly different from the usual and is then used to provide very simply the coefficients of six types of continued fractions whose convergents form ordered sequences of pade approximants including those mentioned above it is then shown that the table of coefficients generated by the revised quotient difference algorithm also provide the partial numerators and denominators of a general continued fraction whose convergents are a sequence of pade approximants that is made up of any chosen horizontal or vertical connected subsequences this freedom in choosing the path in the pade table means that some of the existing algorithms based on particular paths can be regarded as special cases of the revised quotient difference algorithm two such instances are described the revised quotient difference algorithm used in this paper is essentially the algorithm that has already been used to construct the continued fractions whose convergents form the so called two point pade table for two given power series the only difference being the necessity for an additional rule for defining certain coefficients in the absence of a second series however by looking at the case where

the staircase method integrals for periodic reductions of integrable lattice equations we show in full generality that the staircase method papageorgiou et al 1990 phys lett a 147 106 14 quispel et al 1991 physica a 173 243 66 provides integrals for mappings and correspondences obtained as traveling wave reductions of systems of integrable partial difference equations we apply the staircase method to a variety of equations including the korteweg de vries equation the five point bruschi calogero droghei equation the quotient difference qd algorithm and the boussinesq system we show that in all these cases if the staircase method provides r integrals for an n dimensional mapping with then one can introduce q 2r variables which reduce the dimension of the mapping from n to q these dimension reducing variables are obtained as joint invariants of k symmetries of the mappings our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable we also study reductions on other quad graphs than the regular lattice and we prove linear growth of the multi valuedness of iterates of high dimensional correspondences obtained as reductions of the qd algorithm

a novel nth order difference equation that may be integrable we derive an nth order difference equation as a dual of a very simple periodic equation and construct n 1 2 explicit integrals and integrating factors of this equation in terms of multi sums of products we also present a generating function for the degrees of its iterates exhibiting polynomial growth in conclusion we demonstrate how the equation in question arises as a reduction of a system of lattice equations related to an integrable equation of levi and yamilov these three facts combine to suggest the integrability of the nth order difference equation

discrete painleve equations and their lax pairs as reductions of integrable lattice equations we describe a method to obtain lax pairs for periodic reductions of a rather general class of integrable non autonomous lattice equations the method is applied to obtain reductions of the non autonomous discrete korteweg de vries equation and non autonomous discrete schwarzian korteweg de vries equation which yield a discrete analogue of the fourth painleve equation a q analogue of the sixth painleve equation and the q painleve equation with a symmetry group of affine weyl type e 1 6

on some integrable discrete time systems associated with the bogoyavlensky lattices abstract a new class of integrable lattice systems is introduced which are the time discretisations of the bogoyavlensky systems finite dimensional reductions of these systems are considered that give rise to integrable mappings furthermore the similarity reduction is shown to lead to higher order q difference generalisations of the discrete painleve i equation

initial value problems for lattice equations we describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils in finitely many directions we arrive at discrete goursat problems and in the remaining directions we find cauchy problems next we consider s1 s2 periodic initial value problems in the goursat directions the periodic solutions are generated by correspondences in the cauchy directions assuming the equation to be multi linear the periodic solution can be obtained uniquely by iteration of a particularly simple mapping whose dimension is a piecewise linear function of s1 s2

closed form expressions for integrals of traveling wave reductions of integrable lattice equations we give a method to calculate closed form expressions in terms of multi sums of products for integrals of ordinary difference equations which are obtained as traveling wave reductions of integrable partial difference equations important ingredients are the staircase method a non commutative vieta formula and certain splittings of the lax matrices the method is applied to all equations of the adler bobenko suris classification with the exception of q4

markov type fuzzy decision processes with a discounted reward on a closed interval abstract we formulate a new multi stage decision process with markov type fuzzy transition which is termed markov type fuzzy decision process in the general framework of the decision process both of state and action are assumed to be fuzzy itself the transition of states is defined using the fuzzy relation with markov property and the discounted total reward is described as a fuzzy number on a closed bounded interval to discuss the optimization problem a partial order of convex fuzzy numbers is introduced in this paper the discounted total reward associated with an admissible stationary policy is characterized by a unique fixed point of the contractive mapping moreover the optimality equation for the fuzzy decision model is derived under some continuity conditions also an illustrated example is given to explain the theoretical results and the computation in the paper

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