discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

exact enumeration study of free energies of interacting polygons and walks in two dimensions we present analyses of substantially extended series for both interacting self avoiding walks isaw and polygons isap on the square lattice we argue that these provide good evidence that the free energies of both linear and ring polymers are equal above the temperature thus extending the application of a theorem of tesi et al to two dimensions below the temperature the conditions of this theorem break down in contradistinction to three dimensions but an analysis of the ratio of the partition functions for isap and isaw indicates that the free energies are in fact equal at all temperatures within at least any perceived difference can be interpreted as the difference in the size of corrections to scaling in both problems this may be used to explain the vastly different values of the crossover exponent previously estimated for isap to that predicted theoretically and numerically confirmed for isaw an analysis of newly extended neighbour avoiding self avoiding walk series is also given

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

the commutativity of quantized first and higher order hamiltonians the key point of the hamiltonian formalism of toda molecules is the commutativity of the hamiltonians tr yk tr yl 0 where y in gl n and is a poisson bracket associated with the classical r matrix to quantize the toda molecule we have to consider the q analogue of the above formula in this paper we show the commutativity of the quantized first and higher order hamiltonians trq xm trq x 0 where x is a matrix of quantum group glq n

homological properties of modules over semigroup algebras abstract let s be a semigroup in this paper we investigate the injectivity of l 1 s as a banach right module over l 1 s for weakly cancellative s this is the same as studying the flatness of the predual left module c 0 s for such semigroups s we also investigate the projectivity of c 0 s we prove that for many semigroups s for which the banach algebra l 1 s is non amenable the l 1 s module l 1 s is not injective the main result about the projectivity of c 0 s states that for a weakly cancellative inverse semigroup s c 0 s is projective if and only if s is finite

value iteration and rolling plans for markov control processes with unbounded rewards abstract this paper studies the convergence of value iteration functions and the existence of error bounds for rolling horizon procedures in discrete time markov control processes with borel state and control spaces and unbounded reward functions with a discount factor

adsorption and collapse of self avoiding walks in three dimensions a monte carlo study we consider self avoiding walks on the simple cubic lattice with a unit degree vertex at the origin and confined to the half space in addition the walks interact with the plane z 0 and have a vertex vertex interaction within the walk so that the walk can adsorb at the surface z 0 and also undergo collapse we investigate the interaction between collapse and adsorption using monte carlo methods and explore the form of the phase diagram

a remark on symmetry of stochastic dynamical systems and their conserved quantities the symmetry properties of stochastic dynamical systems described by a stochastic differential equation of stratonovich type and related conserved quantities are discussed extending previous results by misawa new conserved quantities are given by applying symmetry operators to known conserved quantities some detailed examples are presented

casorati determinant solutions for the discrete painleve ii equation we present a class of solutions to the discrete painleve ii equation for particular values of its parameters it is shown that these solutions can be expressed in terms of casorati determinants whose entries are discrete airy functions the analogy between the tau function for the discrete pii and that of the discrete toda molecule equation is pointed out

transfer matrix study of the adsorption of a flexible self avoiding polymer chain in two dimensions the authors model the polymer as a self avoiding walk on a semi infinite square lattice with an energy contribution in from each step along the lattice boundary the critical adsorption temperature ta and the ordinary and special surface exponents of the walk above and at the adsorption temperature respectively are estimated with transfer matrix finite size scaling methods for the adsorption temperature they obtain exp in kta 2 041 or 0 002 and for the crossover and susceptibility exponents of the special transition phi sp 0 501 or 0 003 and gamma 1sp 1 454 or 0 004

constructing solutions to the ultradiscrete painleve equations we investigate the nature of particular solutions to the ultradiscrete painleve equations we start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions for the ultradiscrete painleve equations ii and iii we present special solutions reminiscent of the casorati determinant ones which exist in the continuous and discrete cases finally we analyse the discrete painleve equation i and show how it contains both the continuous and the ultradiscrete ones as particular limits

study of the θ point by enumeration of self avoiding walks on the triangular lattice the author reports series expansion analyses of self avoiding walks with nearest neighbour bond interactions the estimates 2 nu t 1 07 or 0 05 and phi 0 64 or 0 05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end to end distance data up to 16 steps on the triangular lattice

crossover exponent for piecewise directed walk adsorption on sierpinski fractals we study the problem of critical adsorption of piecewise directed random walks on a boundary of fractal lattices that belong to the sierpinski gasket family by applying the exact real space renormalization group method we calculate the crossover exponent associated with the number of adsorbed steps for the complete fractal family we demonstrate that our results are very close to the results obtained for ordinary self avoiding walk and discuss the asymptotic behaviour of at the fractal to euclidean lattice crossover

special solutions of discrete painleve equations through direct linearisation abstract we study the special solutions of discrete painleve equations which can be expressed through the discrete equivalent of quadratures the discrete painleve equations examined are the ones which appear in the classification based on affine weyl groups and which can be written as a system of two first order mappings the solutions obtained are the discrete equivalents of solutions which have been shown to exist for the continuous painleve equations

on the construction of approximate solutions for a multidimensional nonlinear heat equation we study three methods based on continuous symmetries to find approximate solutions for the multidimensional nonlinear heat equation delta u delta x0 delta u aun epsilon f u where a and n are arbitrary real constants f is a smooth function and 0 epsilon 1

q deformed oscillator associated with the calogero model and its q coherent state we present the q deformed para bose oscillators associated with the two body calogero model the q deformed coherent state is also constructed and its resolution of unity is demonstrated

chain polymers near an adsorbing surface we present improved estimates of critical exponents for 3d polymers near a plane adsorbing surface the improvements were possible by simulating rather long chains up to n 2000 by means of a new algorithm apart from minor but significant adjustments in the location of the tricritical point and in the various gamma exponents our main result is that the crossover exponent for critical adsorption is compatible with phi 1 2 this is much lower than previous estimates and suggests that this exponent might be superuniversal

a realization of generalized deformed oscillators with cyclic symmetry the algebra of generalized deformed oscillators that possess cyclic symmetry is studied the general set of multilinear operator relations is presented which characterizes the multilinear part of the algebra of cyclic extended deformed oscillators the realization scheme of generalized deformed oscillators with cyclic symmetry in terms of those without cyclic symmetry is established it is shown that all generalized deformed oscillators with cyclic symmetry constructed by the scheme are actually equivalent in the viewpoint of mathematics though each generalized deformed oscillator admits its own particular cyclic extended counterpart examples are given to illustrate this peculiar realization scheme

comment on a theorem of hojman and its generalizations an invariant derived using hojman s conservation law is shown to be trivial for all noether symmetries

yang baxterization of reflection equations for r with two distinct eigenvalues a condition of yang baxterization of reflection equations for r with two distinct eigenvalues is given constant solutions to reflection equations for r related to fundamental representations of slq n and slq 1 1 and for r of the eight vertex model are yang baxterized

special functions as solutions to discrete painleve equations we present results on special solutions of discrete painleve equations these solutions exist only when one constraint among the parameters of the equation is satisfied and are obtained through the solutions of linear second order discrete equations these linear equations define the discrete analogues of special functions

surface critical exponents for models of polymer collapse and adsorption the universality of the theta and theta points the surface critical exponents are estimated both in ordinary and in special regimes from exact series expansions of up to 28 terms for a model of the theta point they are found to be in agreement with those derived for the theta point confirming the conjecture that the theta and theta points lie in the same universality class some essential features of the phase diagram for a self interacting self avoiding polymer in solution are also calculated to a higher degree of accuracy than previously obtained

the effect of attractive monomer monomer interactions on adsorption of a polymer chain the authors have used transfer matrix techniques to study a model of polymer adsorption and collapse they consider the conventional model of a self avoiding walk with attractive nearest neighbour interactions on the square lattice and introduce adsorbing boundaries the numerical results are consistent with the existence of a multicritical point where the adsorption and collapse transitions coincide and suggest that the adsorption transition of a collapsed polymer chain in first order there is also clear evidence that the collapse transition lowers the adsorption temperature estimates of the surface scaling dimensions at the ordinary theta transition support the conjecture of seno and stella 1988 that the surface critical behaviour at the theta and theta points belong to different universality classes

adsorption of self avoiding walks at an impenetrable plane in the expanded phase a monte carlo study we consider self avoiding walks on the simple cubic lattice confined to the half space z0 in addition the walks interact with the plane z 0 and have a vertex vertex interaction the walks can adsorb at the plane z 0 the location of the adsorption phase transition depends in general on the contact interaction we use monte carlo methods to investigate the form of the adsorption phase boundary in the expanded phase region

classical dynamical origin of feynman paths even though only the classical action explicitly appears in the feynman path integral quantum fluctuations show up in the fluctuating nature of feynman paths this prevents a classical interpretation of the latter objects still it makes sense to ask whether the feynman paths have any relation to the classical dynamical trajectories in configuration space a possibility naturally suggested by the semiclassical approximation unfortunately no such relation exists in general things are quite different for the generalized feynman paths which actually contribute in a novel path integral giving an alternative representation of the quantum mechanical propagator it is shown that these paths fluctuate about a classical dynamical trajectory in configuration space because of a certain white noise that is to say the generalized feynman paths are fluctuating curves that collapse on a classical dynamical trajectory as h to 0 this result throws new light on the interplay between classical and quantum mechanics and hints at a new approach to the quantum theory

from continuous painleve iv to the asymmetric discrete painleve i we examine the discrete equation that is obtained from the schlesinger transformations of the continuous painleve iv equation the use of the schlesinger transformations naturally establishes the lax pair of the discrete equation which is an asymmetric form of d p and has as continuous limit we analyse the transformations of the discrete equation and show that they lead to a self dual description where the discrete equation and the schlesinger s play the same role finally using the relation between and asymmetric d we examine in detail the special solutions of both equations

singularity confinement and full deautonomisation a discrete integrability criterion abstract we present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector our method is based on the full deautonomisation procedure which consists in analysing non autonomous extensions of a given discrete system obtained by adding terms that are initially absent but whose presence does not alter the singularity pattern a justification for this approach is given through an algebro geometric analysis we also introduce the notions of early and late confinement while the former is a confinement that may exist already for the autonomous system the latter corresponds to a singularity pattern longer than that of the autonomous case late confinement will be shown to play an important role in the singularity analysis of systems with non trivial gauge freedom for which the existence of an undetected gauge in conjunction with a sketchy analysis might lead to erroneous conclusions as to their integrability an algebro geometric analysis of the role of late confinement in this context is also offered this novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping

recurrent construction of q boson realizations of quantum matrix element algebras of quantum group gl n q the q boson realizations of the quantum matrix element algebra a n q of the quantum group gl n q are recurrently constructed generators of a n q are expressed as n 1 independent q boson operators and the generators of a n 1 q

robustness inequality for markov control processes with unbounded costs the paper discusses the robustness of discrete time markov control processes whose transition probabilities are known up to certain degree of accuracy upper bounds of increase of a discounted cost are derived when using an optimal control policy of the approximating process in order to control the original one bounds are given in terms of weighted total variation distance between transition probabilities they hold for processes on borel spaces with unbounded one stage costs functions

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

coalescence cascades and special function solutions for the continuous and discrete painleve equations we present a systematization of the special function type solutions of continuous and discrete painleve equations our method is to start from and construct its special solutions from the solutions of the hypergeometric equation and by coalescence obtain the lower painleve equations together with their special solutions in the discrete case we study the symmetric one component three point mapping forms of discrete painleve equations starting from d

equivalence transformations and approximate solutions of a nonlinear heat conduction model we propose a method of finding approximate solutions associated with nonlinear partial differential equations involving a small parameter by using equivalence transformations as an application of our method we consider a one dimensional nonlinear heat conduction model and study its approximate solutions

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

miura transforms for discrete painleve equations a miura transformation is presented relating the discrete painleve ii equation to what plays the role of the modified d pii equation 34 in the painleve gambier classification the study of this transformation makes possible the derivation of the auto backlund transform of the discrete pii and allows one to obtain particular solutions to the latter moreover the use of this transformation makes it possible to construct a new mapping in the parameter space of pii the latter is a novel method for the construction of integrable discrete systems

dynamics of the q deformed lipkin model in this paper we investigate the dynamics of the lipkin meshkov glick model within the context of quantum algebras for this purpose we obtain the equations of motion with the help of the time dependent variational principle to analyse the results graphs for equi energies potentials and mean square deviation of the jz operator have been plotted modifications in the usual libration and rotational motions due to the introduction of quantum deformation are discussed

boundary critical behavior of d 2 self avoiding walks on correlated and uncorrelated vacancies in this paper we present exact results for the critical exponents of interacting self avoiding walks with ends at a linear boundary effective interactions are mediated by vacancies correlated and uncorrelated on the dual lattice by choosing different boundary conditions several ordinary and special regimes can be described in terms of clusters geometry and of critical and lowtemperature properties of the mathcal o n 1 model in particular the problem of boundary exponents at the point is fully solved and implications for point universality are discussed the surface crossover exponent at the special transition of noninteracting self avoiding walks is also interpreted in terms of percolation dimensions

the discrete painleve ii equations miura and auto backlund transformations we present miura transformations for all discrete painleve ii equations known to date we then use these miuras to derive special solutions in terms of discrete airy functions and to construct auto backlund transformations for the discrete painleve equations these transformations are then used to generate rational solutions some new forms of d pii and d p34 are obtained as well

dynamics of surface roughening in disordered media the authors present results on the roughening of growing interfaces obtained from a kardar parisi zhang kpz type continuum equation with quenched additive noise representing frozen in disorder close to the pinning transition for the exponents describing respectively the temporal and the spatial scaling of the surface from numerical integration in 1 1 dimensions they obtain beta 0 61 or 0 06 and alpha 0 71 or 0 08 up to a crossover time these estimates are in good agreement with the theoretical prediction beta 3 5 and alpha 3 4 they derive from a dimensional analysis of the equation

adsorption and collapse of self avoiding walks at a defect plane we consider self avoiding walks and polygons on the simple cubic lattice with a vertex vertex interaction i e a contact interaction which also interact with a defect plane z 0 the walks or polygons can undergo a collapse transition and can also adsorb at the defect plane for the polygon case we prove that the limiting free energy exists and show that there is an adsorption transition for each finite value of the vertex vertex interaction we use monte carlo methods to investigate the form of the phase diagram for the walk problem

localized excitations of the 2 1 dimensional sine gordon system the 2 1 dimensional sine gordon system is investigated via a generalized bilinear operator representation plateau basin bowl and saddle type ring solitons are thereby constructed it is indicated how iterated moutard transformations may be employed to extend the range of the method

adsorbing and collapsing trees self interacting branched polymers interacting with a wall can be modelled by lattice trees in a half space with a nearest neighbour contact fugacity and a visit fugacity conjugate to the number of visits the tree makes to the wall which is the boundary of the half space we show that there is a limiting free energy in this model and that it is a non analytic function of the visit fugacity for every finite and fixed value of the contact fugacity this implies the existence of an adsorption transition in this model and there is a critical curve in the phase diagram which separates the phase of desorbed trees from a phase of adsorbed trees moreover we show that for all finite values of the contact fugacity and the adsorption transition occurs at a strictly attractive value of the interaction between the tree and the wall

the quantum group glh 2 among all quantum group structures on the space mat 2 of 2 by 2 matrices how many have a central quantum determinant so that one can define quantum sl 2 out of quantum gl 2 up to isomorphism there are two such structures glq 2 and glh 2 the former is well known the latter is described in this letter

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

non commutative q painleve vi equation by applying suitable centrality conditions to non commutative non isospectral lattice modified gel fand dikii type systems we obtain the corresponding non autonomous equations then we derive the non commutative q discrete painleve vi equation with full range of parameters as the 2 2 similarity reduction of the non commutative non isospectral and non autonomous lattice modified korteweg de vries equation we also comment on the fact that in making the analogous reduction starting from the schwarzian korteweg de vries equation no such non isospectral generalization is needed

conservation laws for divergenceless differential equations a paper by hojman 1992 shows how to use a symmetry to construct a constant of the motion for a system of second order differential equations directly from the equations without using either a lagrangian or hamiltonian it is however necessary to impose some constraint on the nature of these equations this constraint is recognized as saying that the differential equation field is divergenceless with respect to some connection on the extended tangent bundle however for the simplest example of a lagrangian consisting of a kinetic energy and velocity independent potential all the conserved quantities constructed in this way from noether symmetries are identically zero the relation of this work to earlier work is pointed out

perturbation and stability theory for markov control problems a unified approach to the asymptotic analysis of a markov decision process disturbed by an epsilon additive perturbation is proposed irrespective of whether the perturbation is regular or singular the underlying control problem that needs to be understood is the limit markov control problem the properties of this problem are studied

new airy type solutions of the ultradiscrete painleve ii equation with parity variables the q difference painleve ii equation admits special solutions written in terms of determinants whose entries are the general solution of the q airy equation an ultradiscrete limit of the special solutions is studied using the procedure of ultradiscretization with parity variables then we obtain new airy type solutions of the ultradiscrete painleve ii equation with parity variables and the solutions have a richer structure than the known solutions

on a class of special solutions of the painleve equations abstract we present a class of special solutions for continuous discrete and q discrete painleve equations these solutions are expressed through quadratures or their discrete equivalents the condition for their existence is a further constraint on the conditions for the existence of other special linearisable solutions of painleve equations thus these solutions exist for equations that have at least two parameters in most cases the special solutions are expressed in terms of functions defined through quadratures

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

do ultradiscrete systems with parity variables satisfy the singularity confinement criterion ultradiscrete singularity confinement test which is an integrability detector for ultradiscrete equations with parity variables is applied to various ultradiscrete equations the ultradiscrete equations exhibit singularity structures analogous to those of the discrete counterparts exact solutions to linearisable ultradiscrete equations are also constructed to explain the singularity structures

casorati determinant solutions for the discrete painleve iii equation the discrete painleve iii equation is investigated based on the bilinear formalism it is shown that it admits the solutions expressed by the casorati determinant whose entries are given by the discrete bessel functions moreover based on the observation that these discrete bessel functions are transformed to the q bessel functions by a simple variable transformation we present a q difference analog of the painleve iii equation

limiting optimal discounted cost control of a class of time varying stochastic systems abstract we consider a class of time varying r d valued control models and with possibly unbounded costs the processes evolve according to the system equation x n 1 g n x n a n n n n where n are i i d random vectors and g n a sequence of known functions converging to some function g under suitable hypotheses we show the existence of an discount optimal policy for the limiting system x n 1 g x n a n n

a corner transfer matrix renormalization group investigation of the vertex interacting self avoiding walk model ar ecently introduced extension of the corner transfer matrix renormalization group method useful for the study of self avoiding walk type models is presented in detail and applied to a class of interacting self avoiding walks due to bl ote and nienhuis this model displays two different types of collapse transition depending on model parameters one is the standard point transition the other is found to give rise to a first order collapse transition despite being known to be in other respects critical

self interacting polymer chains terminally anchored to adsorbing surfaces of three dimensional fractal lattices abstract we have studied the adsorption problem of self attracting linear polymers modeled by self avoiding walks saws situated on three dimensional fractal structures exemplified by 3d sierpinski gasket sg family of fractals as containers of a poor solvent members of sg family are enumerated by an integer b b 2 and it is assumed that one side of each sg fractal is an impenetrable adsorbing surface we calculate the critical exponents 1 11 and s which are related to the numbers of all possible saws with one both and no ends anchored to the adsorbing boundary respectively by applying the exact renormalization group rg method for the first three members of the sg fractal family b 2 3 and 4 we have obtained specific values of these exponents for chain and globular polymer phase we discuss their mutual relations and relations with corresponding values pertinent to extended polymer chain phase

determinant structure of the rational solutions for the painleve iv equation rational solutions for the painleve iv equation are investigated by hirota bilinear formalism it is shown that the solutions in one hierarchy are expressed by 3 reduced schur functions and those in another two hierarchies by casorati determinants of the hermite polynomials or by a special case of the schur polynomials

special function solutions for asymmetric discrete painleve equations we study the solutions of a family of discrete painleve equations the equations that we examine are given as a system of two first order non autonomous mappings the solutions we are interested in are the ones obtained whenever the painleve equation can be reduced to a discrete riccati equation which can be linearized through a cole hopf transformation the special solutions thus obtained involve generalizations or reductions of the hypergeometric and q hypergeometric function

flatigw an inverse algorithm to compute the density of states of lattice self avoiding walks we show that the density of states dos for lattice self avoiding walks can be estimated by using an inverse algorithm called flatigw whose step growth rules are dynamically adjusted by requiring the energy histogram to be locally flat here the attractive energy associated with a configuration is taken to be proportional to the number of non bonded nearest neighbor pairs contacts the energy histogram is able to explicitly direct the growth of a walk because the step growth rule of the interacting growth walk narasimhan et al 2003 5 samples the available nearest neighbor sites according to the number of contacts they would make we have obtained the complex fisher zeros corresponding to the dos estimated for square lattice walks of various lengths and located the temperature by extrapolating the finite size values of the real zeros to their asymptotic value 1 49 reasonably close to the known value 1 50 barkema et al 1998 14

q boson realization of quadratic algebra a1 and its representations the non generic central elements of the quadratic algebra a1 associated with the quantum group gl 2 q are found in the case where q is a root of unity a q boson realization of a1 is constructed in terms of the q boson realization the representations of a1 on the q fock space are studied in both generic and non generic cases and the cyclic representation is obtained in the non generic case

determinant structure of the rational solutions for the painleve ii equation two types of determinant representations of the rational solutions for the painleve ii equation are discussed by using the bilinear formalism one of them is a representation by the devisme polynomials and another one is hankel determinant representation they are derived from the determinant solutions of the kp hierarchy and toda lattice respectively

adsorption and collapse of self avoiding walks and polygons in three dimensions we consider self avoiding walks and polygons on the simple cubic lattice confined to the half space and interacting with the plane z 0 in addition there is a short range vertex vertex interaction in the walk or polygon which can lead to a collapse transition we explore the interaction between collapse and adsorption in these systems and discuss the form of the phase diagram key results include a proof of the existence of an adsorption transition for polygons for every value of the vertex vertex interaction a corresponding proof for walks when the vertex vertex interaction term is repulsive and a proof that if polygons exhibit a collapse transition then the phase boundary between the expanded and desorbed phase and the collapsed and desorbed phase must be a straight line

a robustness result for stochastic control abstract the solution of a stochastic control problem depends on the underlying model the actual real world model may not be known precisely and so one solves the problem for a hypothetical model that is in general different but close to the real one the optimal or nearly optimal control of the hypothetical model is then used as solution for the real problem in this paper we assume that what is not precisely known is the underlying probability measure that determines the distribution of the random quantities driving the model we investigate two ways to derive a bound on the suboptimality of the optimal control of the hypothetical problem when this control is used in the real problem both bounds are in terms of the radon nikodym derivative of the underlying real world measure with respect to the hypothetical one we finally investigate how the bounds compare to each other

exact enumeration results for self avoiding walks on the honeycomb lattice attached to a surface we consider self avoiding walks on the honeycomb lattice interacting with a surface with different energies associated between sites in contact with a linear boundary to the left of the origin and those in contact with the right of the boundary we numerically confirm recent exact results for the polymer adsorption transition and corresponding critical exponents with mixed ordinary and special boundary conditions the phase diagram is elucidated with the aid of some rigorous arguments

special function solutions of the discrete painleve equations in this paper we present a review of the special solutions of discrete difference and q discrete painleve equations in terms of discrete special functions these solutions exist whenever the parameters of the painleve equation satisfy a particular constraint the discrete special functions belong to the hypergeometric family although in some cases they seem to go beyond the known discrete and q discrete hypergeometric functions the equations studied in this paper are chosen on the basis of our recent classification of discrete painleve equation with the help of affine weyl groups

homological properties of modules over group algebras let g be a locally compact group and let l 1 g be the banach algebra which is the group algebra of g we consider a variety of banach left l 1 g modules over l 1 g and seek to determine conditions on g that determine when these modules are either projective or injective or flat in the category the answers typically involve g being compact or discrete or amenable for example in the case where g is discrete and 1 p infty we find that the module ell p g is injective whenever g is amenable and that if it is amenable then g is pseudo amenable a property very close to that of amenability

on some aspects of backlund transformations the painleve differential equations p 2 p 6 possess backlund transformations which relate one solution to another solution either of the same equation with different values of the parameters or another such equation we review a method for deriving difference equations the discrete painleve equations in particular from backlund transformations of the continuous painleve equations then we prove the existence of an algebraic formula relating three inconsecutive solutions of the same backlund hierarchy for p 3 and p 4

the discrete first second and thirty fourth painleve hierarchies the discrete first and second painleve equations dp and dp are integrable difference equations which have classical first second or third painleve equations p p or p as continuum limits dp and dp are believed to be integrable because they are discrete isomonodromy conditions for associated single valued linear problems an infinite hierarchy of integrable difference equations that share the same linear deformation problem as dp was shown to exist by cresswell and joshi in this paper we recall the results shown for dp and show how to deduce a hierarchy for dp each member of the respective hierarchies is shown to be generated by difference recursion operators furthermore we show that continuum limits of these difference hierarchies lead to the p p and p hierarchies finally we construct miura transformations of the dp hierarchy and show that these lead to the hierarchy of the discrete thirty fourth painleve equation

the q painleve v equation and its geometrical description we study the q painleve v equation which can be obtained from the degeneration of the q pvi in the form of the asymmetric q piii equation and present its geometrical description based on the bilinear formulation we obtain the equations for the multi dimensional functions of q pv in the form of nonautonomous hirota miwa systems which lives in the weight lattice of the a4 affine weyl group this geometrical approach furnishes in a straightforward way the miuras and the schlesingers of q pv

deformation of the strange superalgebra p n a deformation uq p n pq n of the extended noncontragredient strange superalgebra of p n denoted by p n is proposed a realization of pq n in terms of q bosons and q fermions is presented then a procedure to build up a set of representations of pq n is briefly discussed as a by product a new realization of suq n is obtained

bilinear discrete painleve ii and its particular solutions by analogy to the continuous painleve ii equation we present particular solutions of the discrete painleve ii d pii equation these solutions are of a rational and special function airy type our analysis is based on the bilinear formalism that allows us to obtain the tau function for d pii two different forms of bilinear d pii are obtained and we show that they can be related by a simple gauge transformation