numerical approximation for functionals of reflecting diffusion processes the aim of this paper is to approximate the expectation of a large class of functionals of the solution x xi of a stochastic differential equation with normal reflection in a piecewise smooth domain of rd this also yields a monte carlo method for solving partial differential problems of parabolic type with mixed boundary conditions the approximation is based on a modified euler scheme for the stochastic differential equation the scheme can be driven by a sequence of bounded independently and identically distributed i i d random variables or when the domain is convex by a sequence of gaussian i i d random variables the order of weak convergence for both cases is given in the former case the order of convergence is 1 2 and it is shown to be exact by an example in the last section numerical tests are presented the behavior of the error as a function of the final time t for fixed values of the discretization step and as a function of the discretization step for fixed values of the final

some analytical consequences of the inverse relation for the potts model the authors present some analytical results for the potts model using the symmetry group acting on its parameters generated by the inverse relation and other symmetries of this model in particular they find the critical manifolds and study the relationship between this group and the lee yang singularities in the complex plane for those cases where the symmetry group is finite they look at the possible consequences for some colouring problems in graph theory and more specifically for chromatic polynomials

a splitting step algorithm for reflected stochastic differential equations in r 1 a new algorithm which is based on the splitting step idea and the penalization method for reflected stochastic differential equation rsde in the upper half space r 1 is presented in this paper after some important estimates about rsdes and penalization odes are obtained the local pathwise error of the suggested algorithm are considered and an estimate of local error in the 2pth moment is proved under the assumptions of lipschitz continuity and linear growth some numerical experiments based on matlab programs are given to support the suggested algorithm

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

existence uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching we study viscosity solutions to a system of nonlinear degenerate parabolic partial integrodi erential equations with interconnected obstacles this type of problem occurs in the context of optimal

reflected solutions of backward sde s and related obstacle problems for pde s we study reflected solutions of one dimensional backward stochastic differential equations the reflection keeps the solution above a given stochastic process we prove uniqueness and existence both by a fixed point argument and by approximation via penalization we show that when the coefficient has a special form then the solution of our problem is the value function of a mixed optimal stopping optimal stochastic control problem we finally show that when put in a markovian framework the solution of our reflected bsde provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation

stochastic control representations for penalized backward stochastic differential equations this paper shows that the penalized backward stochastic differential equation bsde which is often used to approximate and solve the corresponding reflected bsde admits both optimal stopping representation and optimal control representation the new feature of the optimal stopping representation is that the player is allowed to stop at exogenous poisson arrival times the convergence rate of the penalized bsde then follows from the optimal stopping representation the paper applies to two classes of equations namely multidimensional reflected bsde and reflected bsde with a constraint on the hedging part and gives stochastic control representations for their corresponding penalized equations

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

hyperbolic coxeter groups for triangular potts models the symmetry groups generated by the inversion relations of lattice models of statistical mechanics on triangular lattices are analysed for vertex models and for the standard scalar potts model with two and three site interactions these groups are generated by three inversion relations and are seen to be generically very large ones hyperbolic groups two situations for which the representations of these groups degenerate into smaller ones hopefully compatible with integrability are considered the first reduction for the vertex triangular model corresponds to the situation where the vertex of the triangular model coincides with the left or right hand side of a yang baxter relation in this case the representation of the group is isomorphic up to a semi direct product by a finite group to z z the second reduction for q state ports models occurs for particular values of q the so called tutte beraha numbers for this model algebraic varieties including the known ferromagnetic critical variety happen to be invariant under such large groups of symmetries as a byproduct this analysis provides nice birational representations of hyperbolic coxeter groups

the inversion relation method for some two dimensional exactly solved models in lattice statistics the partition functions per site of several two dimensional models notably the eight vertex self dual potts and hard hexagon models can be easily obtained by using an inversion relation for local transfer matrices together with symmetry and analyticity properties this technique is discussed the analyticity properties compared and some equivalences and nonequivalences pointed out in particular the critical hard hexagon model is found to have the same as the self dualq state potts model withq 3 5 2 2 618 the temperley lieb equivalence between the potts and six vertex models is found to fail in certain nonphysical antiferromagnetic cases

on the equality of solutions of max min and min max systems of variational inequalities with interconnected bilateral obstacles abstract in this paper we deal with the solutions of systems of pdes with bilateral interconnected obstacles of min max and max min types these systems arise naturally in stochastic switching zero sum game problems we show that when the switching costs of one side are regular the solutions of the min max and max min systems coincide then this common viscosity solution is related to a multi dimensional doubly reflected bsde with bilateral interconnected obstacles finally its relationship with the values of a zero sum switching game is studied

rate of convergeance of a particle method for the solution of a 1d viscous scalar conservation law in a bounded interval in this paper we give a probabilistic interpretation of a viscous scalar conservation law in a bounded interval thanks to a nonlinear martingale problem the underlying nonlinear stochastic process is reflected at the boundary to take into account the dirichlet conditions after proving uniqueness for the martingale problem we show existence thanks to a propagation of chaos result indeed we exhibit a system of n interacting particles the empirical measure of which converges to the unique solution of the martingale problem as n as a consequence the solution of the viscous conservation law can be approximated thanks to a numerical algorithm based on the simulation of the particle system when this system is discretized in time thanks to the euler lepingle scheme d lepingle math comput simulation 38 1995 119 126 we show that the rate of convergence of the error is in o t 1 n where at denotes the time step finally we give numerical results which confirm this theoretical rate

dynkin games via dirichlet forms and singular control of one dimensional diffusions we consider a zero sum game of optimal stopping in which each of the opponents has the right to stop a one dimensional diffusion process there are two types of costs the first is accumulated continuously at the rate h xt where xt is the current position of the process the second is a cost associated with the stopping of the process it is given by the function f1 x for the first player and the function f2 x for the second player where x is the position of the process when the stopping option is exercised we study the solution of the free boundary problem associated with this game via dirichlet forms on the appropriate functional space integrating the value function of the game we get a solution to another free boundary problem which yields the optimal return function for a singular stochastic control problem

an explicit formula for the skorokhod map on 0 a the skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions in this work an explicit formula for the skorokhod map 0 a on 0 a for any a 0 is derived specifically it is shown that on the space d 0 of right continuous functions with left limits taking values in r 0 a a o 0 where a d 0 d 0 is defined by a t t sups 0 t s a infu s t u and 0 d 0 d 0 is the skorokhod map on 0 which is given explicitly by 0 t t sup s 0 t s in addition properties of a are developed and comparison properties of 0 a are established

large q expansion for the anisotropic potts model a large q expansion of the partition function of the anisotropic q state potts model in two dimensions is given up to the sixth order in q 12 at the critical temperature it is in agreement with the exact solutions for the partition function and for the latent heat the latter does not depend on the anisotropy it does not show any simple factorisation in the parameters of the model

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

sdes with oblique reflection on nonsmooth domains in this paper we consider stochastic differential equations with reflecting boundary conditions for domains that might have corners and for which the allowed directions of reflection at a point on the boundary of the domain are possibly oblique the main results are strong existence and uniqueness for solutions of such equations a key ingredient is a family of relatively regular functions appropriate to the given domain and directions of reflection two cases are treated in the paper in the first case the direction of reflection is single valued and varies smoothly and the main new feature is that the boundary of the domain may be nonsmooth in the second case the domain is taken to be the intersection of a finite number of domains with relatively smooth boundary and at the resulting corner points more than one oblique direction is allowed

probabilistic approach for nonlinear partial differential equations and stochastic partial differential equations with neumann boundary conditions abstract in this study we consider nonlinear partial differential equations pdes and stochastic pdes spdes with neumann boundary conditions on domains that satisfy the lions sznitmann saisho conditions using the convergence result based on the penalization approximation for stochastic differential equations with normal reflections on nonsmooth and nonconvex domains we establish the existence and comparison principle for the viscosity solutions of nonlinear pdes with the neumann conditions associated with the optimal control problem we also obtain for nonlinear spdes and backward spdes with neumann condition the representations of the stochastic viscosity solutions

inversion relations and disorder solutions on potts models the inversion symmetry and the automorphy group generated by the latter when combined with the spatial symmetries are studied for the anisotropic potts models on the triangular and checkerboard lattices the exact expression of the partition function which is known on some particular disorder subvarieties of the triangular model is checked and a generalisation is proposed for the checkerboard lattice the automorphy group is then used to extend the disorder solutions to the infinity of transformed subvarieties

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

fluctuations for a φ interface model with repulsion from a wall we consider a interface model on a one dimensional lattice with repulsion from a hard wall we suppose that the repulsion is of the form c 1 where c 0 and denotes the height of the interface from the wall we prove convergence of the equilibrium fluctuations around the hydrodynamic limit to the solution of a spde with singular drift if c 0 the system becomes the funaki olla interface model with reflection at the wall whose equilibrium fluctuations converge to the solution of a spde with reflection we give a new proof of this result using the characterization of such solution as the diffusion generated by an infinite dimensional dirichlet form obtained in a previous paper our method is based on a study of integration by parts formulae w r t the equilibrium measure of the interface model and allows to avoid the proof of the so called boltzmann gibbs principle we also obtain convergence of finite dimensional distributions of non equilibrium fluctuations around the stationary hydrodynamic limit 0

multi dimensional bsde with oblique reflection and optimal switching in this paper we study a multi dimensional backward stochastic differential equation bsde with oblique reflection which is a bsde reflected on the boundary of a special unbounded convex domain along an oblique direction and which arises naturally in the study of optimal switching problem the existence of the adapted solution is obtained by the penalization method the monotone convergence and the a priori estimates the uniqueness is obtained by a verification method the first component of any adapted solution is shown to be the vector value of a switching problem for bsdes as applications we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities

classifying regular languages by a split game in this paper we introduce the split game a variant of the ehrenfeucht fraisse game from logic which is useful for analyzing the expressive power of classes of generalized regular expressions an extension of the split game to generalized w regular expressions is also established to gain insight into how the split game can be applied to attack the long standing generalized star height 2 problem we propose and solve the omega power problem a similar but tractable problem in the context of w languages namely we show that omega powers together with boolean combinations and concatenations are not sufficient to express the class of w regular languages

algebraic invariants in soluble models exhibits on the known solutions of the triangle relation for the exactly solved models some simplifying methods of recovering their parametrisation in terms of algebraic varieties the relation to the automorphy group generated by the inverse and spatial symmetries of the model is also analysed

remark on the star height problem abstract a method is presented which allows to construct general regular expressions of star height 1 for a class of rather complicated regular events general regular expressions include an operation symbol for complement if events of greater general star height exist which is still open they must be more complex than those accessible by this method an event which seems to be of this kind is suggested at the end of the paper

a stochastic differential game in the orthrant we study a zero sum stochastic differential game in the nonnegative orthrant the state of the system is governed by controlled reflecting diffusions in the nonnegative orthrant we consider discounted and average payoff evaluation criteria we prove the existence of values and optimal strategies for both payoff criteria

reflected brownian motion with skew symmetric data in a polyhedral domain this paper is concerned with the characterization and invariant measures of certain reflected brownian motions rbm s in polyhedral domains the kind of rbm studied here behaves like d dimensional brownian motion with constant drift in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced in harrison williams 9 it is shown that such an rbm can be characterized in terms of a family of submartingales and that it reaches non smooth parts of the boundary with probability zero in 9 a purely analytic problem associated with such an rbm was solved here the exponential form solution obtained in 9 is shown to be the density of an invariant measure for the rbm furthermore if the density is integrable over the polyhedral state space then it yields the unique stationary distribution for the rbm in the proofs of these results a key role is played by a dual process for the rbm and by results in 9 for reflected brownian motions on smooth approximating domains

q state potts model on the checkerboard lattice a large q expansion of the partition function of the checkerboard q state potts model is given up to sixth order in q 12 this expansion is used to discuss the following two points first on the critical manifold the large q expansion agrees up to this order with the minimal solution suggested by the group structure associated with this model secondly the latent heat and some correlation functions are discussed from the point of view of their functional dependence in the parameters

singularity complexity and quasi integrability of rational mappings we investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models by exploiting their nature of birational transformations of projective spaces we give an algorithmic analysis of the structure of invariants of such mappings we discuss some characteristic conditions for their quasi integrability and in particular its links with their singularities in the 2 plane finally we describe some of their propertiesqua dynamical systems making contact with arnol d s notion of complexity and exemplify remarkable behaviours

inversion relations and symmetry groups for potts models on the triangular lattice inversion relations are obtained for the standard scalar q state triangular potts model with two and three spin interactions generalizing previously known results for two spin interaction models it is shown that these inversion relations generate a group of symmetries of the model which is naturally represented in terms of birational transformations in a four dimensional parameter space this group of birational transformations is generically a very large one namely a hyperbolic coxeter group in this framework of very large groups of symmetries a remarkable situation pops out the one for which q corresponds to tutte beraha numbers

heavy traffic limits for the g h2 n m queue we establish heavy traffic stochastic process limits for queue length waiting time and overflow stochastic processes in a class ofg gi n m queueing models withn servers andm extra waiting spaces we let the arrival process be general only requiring that it satisfy a functional central limit theorem to capture the impact of the service time distribution beyond its mean within a markovian framework we consider a special class of service time distributions denoted byh2 which are mixtures of an exponential distribution with probabilityp and a unit point mass at 0 with probability 1 p these service time distributions exhibit relatively high variability having squared coefficients of variation greater than or equal to one as in halfin and whitt 1981 heavy traffic limits for queues with many exponential servers oper res 29 567 588 puhalskii and reiman 2000 the multiclassgi ph n queue in the halfin whitt regime adv appl probab 32 564 595 and garnett mandelbaum and reiman 2002 designing a call center with impatient customers manufacturing service oper management 4 208 227 we consider a sequence of queueing models indexed by the number of servers n and letn tend to infinity along with the traffic intensities nso that v n 1 n for 8 8 to treat finite waiting rooms we letm n v n for 0 8 with the specialh2 service time distribution the limit processes are one dimensional markov processes behaving like diffusion processes with different drift and diffusion functions in two different regions above and below zero we also establish a limit for theg m n m m model having exponential customer abandonments

lifschitz tail on hyperbolic space neumann conditions on considere un espace hyperbolique h de dimension n et l operateur laplacien hyperbolique h etant donne une distribution aleatoire de points a l aide d un processus ponctuel de poisson tel que chaque point se situe dans une boule hyperbolique de rayons on etudie la question suivante si est un petit nombre reel combien y a t il de valeurs propres inferieures a pour l operateur 1 2 h dans une boule b de centre o avec des conditions de neumann sur la frontiere et avec des conditions aux limites de dirichlet sur les boules aleatoires formant des intersections avec b si le rayon de b est suffisamment grand

on the starting and stopping problem application in reversible investments in this work we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process we use backward stochastic differential equations bsdes and snell envelopes finally we give some numerical results

integrable mappings and polynomial growth abstract we describe birational representations of discrete groups generated by involutions having their origin in the theory of exactly solvable vertex models in lattice statistical mechanics these involutions correspond respectively to two kinds of transformations on q q matrices the inversion of the q q matrix and an involutive permutation of the entries of the matrix we concentrate on the case where these permutations are elementary transpositions of two entries in this case the birational transformations fall into six different classes for each class we analyze the factorization properties of the iteration of these transformations these factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties some mappings yield a polynomial growth of the complexity of the iterations for three classes the successive iterates for q 4 actually lie on elliptic curves this analysis also provides examples of integrable mappings in arbitrary dimension even infinite moreover for two classes the homogeneous polynomials are shown to satisfy non trivial non linear recurrences the relations between factorizations of the iterations the existence of recurrences on one or several variables as well as the integrability of the mappings are analyzed

rational dynamical zeta functions for birational transformations we propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables depending on two parameters this conjectured function is a simple rational expression with integer coecients this yields an algebraic value for the topological entropy furthermore the generating function for the arnold complexity is also conjectured to be a rational expression with integer coecients with the same singularities as for the dynamical zeta function this leads at least in this example to an equality between the asymptotic of the arnold complexity and the exponential of the topological entropy we also give a semi numerical method to eectively compute the arnold complexity c 1999 published by elsevier science b v all rights reserved

more languages of generalised star height 1 abstract suppose that membership of a word w in a language l is specified in terms of the number modulo some integer k of occurrences of a fixed word w as a factor of w then the language obtained for any m by adding arbitrary numbers of copies of a m anywhere in the words of l has generalized star height at most 1

continuous dependence results for non linear neumann type boundary value problems abstract we obtain estimates on the continuous dependence on the coefficient for second order non linear degenerate neumann type boundary value problems our results extend previous work of cockburn et al jakobsen and karlsen and gripenberg to problems with more general boundary conditions and domains a new feature here is that we account for the dependence on the boundary conditions as one application of our continuous dependence results we derive for the first time the rate of convergence for the vanishing viscosity method for such problems we also derive new explicit continuous dependence on the coefficients results for problems involving bellman isaacs equations and certain quasilinear equation

on the oblique derivative problem for diffusion processes and diffusion equations with holder continuous coefficients on a c2 domain in a euclidean space we consider the oblique derivative problem for a diffusion equation and assume the coefficients of the diffusion and boundary operators are holder continuous we then prove the uniqueness of diffusion processes and fundamental solutions corresponding to the problem for the purpose obtaining a stochastic representation of some so lutions to the problem plays a key role in our situation a difficulty arises from the absence of a fundamental solution with c2 smoothness up to the boundary it is overcome by showing some stability of a fundamental solution and a dif fusion process respectively under approximation of the domain in particular the stability of the fundamental solution is verified through construction it is done by applying the parametrix method twice to a parametrix with explicit expression

rational mappings arborescent iterations and the symmetries of integrability we describe a class of nonlinear birational representations of groups generated by a finite number of involutions these groups are symmetries of the yang baxter equations and their higher dimensional generalizations they provide discrete dynamical systems with a variety of behaviors from chaotic to integrable according to the number of invariants of the representation

discrete symmetry groups of vertex models in statistical mechanics we analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations they can be seen as generated by involutions corresponding respectively to two kinds of transformations onq q matrices the inversion of theq q matrix and an involutive permutation of the entries of the matrix we show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics this approach enables one to analyze two dimensionalq4 state vertex models as simply as three dimensional vertex models or higher dimensional vertex models various examples of birational symmetries of vertex models are analyzed a particular emphasis is devoted to a three dimensional vertex model the 64 state cubic vertex model which exhibits a polynomial growth of the complexity of the calculations a subcase of this general model is seen to yield integrable recursion relations we also concentrate on a specific two dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes yang baxter integrable it is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces or higher dimensional algebraic varieties

factorization properties of birational mappings abstract we analyse birational mappings generated by transformations on q q matrices which correspond respectively to two kinds of transformations the matrix inversion and a permutation of the entries of the q q matrix remarkable factorization properties emerge for quite general involutive permutations it is shown that factorization properties do exist even for birational transformations associated with noninvolutive permutations of entries of q q matrices and even for more general transformation which are rational transformations but no longer birational the existence of factorization relations independent of q the size of the matrices is underlined the relations between the polynomial growth of the complexity of the iterations the existence of recursions in a single variable and the integrability of the mappings are sketched for the permutations yielding these properties all these results show that permutations of the entries of the matrix yielding factorization properties are not so rare in contrast the occurrence of recursions in a single variable or of the polynomial growth of the complexity are of course less frequent but not completely exceptional

the submartingale problem for a class of degenerate elliptic operators we consider the degenerate elliptic operator acting on c 2 b functions on 0 d mathcal l f x sum i 1 d a i x x i alpha i frac partial 2 f partial x i 2 x sum i 1 d b i x frac partial f partial x i x where the ai are continuous functions that are bounded above and below by positive constants the bi are bounded and measurable and the alpha i in 0 1 we impose neumann boundary conditions on the boundary of 0 d there will not be uniqueness for the submartingale problem corresponding to mathcal l if we consider however only those solutions to the submartingale problem for which the process spends 0 time on the boundary then existence and uniqueness for the submartingale problem for mathcal l holds within this class our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations rm d x t i sqrt 2a i x t x t i alpha i 2 rm d w i t b i x t rm d t rm d l t x i quad x i t geq 0 where w t i are independent brownian motions and l x i t is a local time at 0 for xi

ergodicity of robust switching control and nonlinear system of quasi variational inequalities we analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities these equations are related to robust switching control problems introduced in 3 we prove that as time horizon goes to infinity resp discount factor goes to zero the long run average solution to the parabolic system resp the limiting discounted solution to the elliptic system is characterized by a solution of a nonlinear system of ergodic variational inequalities our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities

growth complexity spectrum of some discrete dynamical systems we first study the iteration of birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of 3 3 matrices and consider the degree d n of the numerators or denominators of the corresponding successive rational expressions for the nth iterate the growth of this degree is generically exponential with n d n n is called the growth complexity we introduce a semi numerical analysis which enables to compute these growth complexities for all the 9 possible birational transformations these growth complexities correspond to a spectrum of eighteen algebraic values we then drastically generalize these results replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters again it is shown that the associated birational or even rational transformations yield algebraic values for their growth complexities 1999 elsevier science b v all rights reserved

robust feedback switching control dynamic programming and viscosity solutions we consider a robust switching control problem the controller only observes the evolution of the state process and thus uses feedback closed loop switching strategies a nonstandard class of switching controls introduced in this paper the adverse player nature chooses open loop controls that represent the so called knightian uncertainty i e misspecifications of the model the half game switcher versus nature is then formulated as a two step robust optimization problem we develop the stochastic perron s method in this framework and prove that it produces a viscosity subsolution and supersolution to a system of hjb variational inequalities which envelop the value function together with a comparison principle this characterizes the value function of the game as the unique viscosity solution to the hjb equation and shows as a by product the dynamic programming principle for the robust feedback switching control problem

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

three dimensional vertex model in statistical mechanics from baxter bazhanov model we find that the boltzmann weight of the three dimensional baxter bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the wu kadanoff wegner duality between the cube and vertex type tetrahedron equations is obtained explicitly for the baxter bazhanov model then a three dimensional vertex model is obtained by considering the symmetry property of the weight function which corresponds to the three dimensional baxter bazhanov model the vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube we write down the symmetry relations of the weight functions under the actions of the symmetry groupg of the cube the six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected with the six spaces in which the vertextype tetrahedron equation is defined

more integrable birational mappings we study birational mappings generated by matrix inversion and permutation of the entries of q q matrices for q 3 we have performed a systematic examination of all the permutations of 3 3 matrices in order to find integrable mappings of three different kinds and finite order mappings this exhaustive analysis gives among 30 462 classes of mappings 27 new integrable classes of birational mappings and 36 classes yielding finite order recursions associated with these mappings an exhaustive analysis with a constraint on the diagonal entries has also been performed for 4 4 matrices we have found 8306 new classes of integrable mappings all these new examples show that integrability can actually correspond to non involutive permutations the analysis of the integrable cases specific of a particular size of the matrix and a careful examination of the non involutive permutations could shed some light on integrability of such birational mappings it seems that one has the following result the non involutive examples are specific of a given matrix size 3 3 matrix and the permutations which yield integrable mappings for arbitrary matrix size are always involutions

backward sdes with constrained jumps and quasi variational inequalities we consider a class of backward stochastic differential equations bsdes driven by brownian motion and poisson random measure and subject to constraints on the jump component we prove the existence and uniqueness of the minimal solution for the bsdes by using a penalization approach moreover we show that under mild conditions the minimal solutions to these constrained bsdes can be characterized as the unique viscosity solution of quasi variational inequalities qvis which leads to a probabilistic representation for solutions to qvis such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems as a direct consequence this suggests a numerical scheme for the solution of such qvis via the simulation of the penalized bsdes

optimal switching over multiple regimes this paper studies the optimal switching problem for a general one dimensional diffusion with multiple more than two regimes this is motivated in the real options literature by the investment problem of a firm managing several production modes while facing uncertainties a viscosity solutions approach is employed to carry out a fine analysis on the associated system of variational inequalities leading to sharp qualitative characterizations of the switching regions these characterizations in turn reduce the switching problem into one of finding a finite number of threshold values in a state that would trigger switchings the results of our analysis take several qualitatively different forms depending on model parameters and the issue of when and where it is optimal to switch is addressed the general results are then demonstrated by the three regime case where a quasi explicit solution is obtained and a numerical procedure to find these critical values is devised in terms of the expectation functionals of hitting times for one dimensional diffusions

automorphisms of algebraic varieties and yang baxter equations it is shown that commuting transfer matrix models in statistical mechanics are parametrized by algebraic varieties having a set of automorphisms deduced from the so called inversion relation in general this set of automorphisms is infinite this shows that for algebraic varieties of dimension 1 the models are parametrized by algebraic curves of genus 0 or 1

some results on the generalized star height problem abstract we prove some results related to the generalized star height problem in this problem as opposed to the restricted star height problem complementation is considered as a basic operator we first show that the class of languages of star height n is closed under certain operations left and right quotients inverse alphabetic morphisms injective star free substitutions it is known that languages recognized by a commutative group are of star height 1 we extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by z 2 z n in the same direction we show that one of the languages that were conjectured to be of star height 2 during the past ten years is in fact of star height 1 next we show that if a rational language l is recognized by a monoid of the variety generated by wreath products of the form m g n where m and n are aperiodic monoids and g is a commutative group the l is of star height 1 finally we show that every rational language is the inverse image under some morphism between free monoids of a language of resticted star height 1

a multidimensional process involving local time on presente une approche probabiliste a un probleme pose par mckean 1967 pour la construction d un systeme de n particules sur r avec un generateur formel

new integrable cases of a cremona transformation a finite order orbits analysis we analyse the properties of a particular birational mapping of two variables cremona transformation depending on two free parameters e and associated with the action of a discrete group of non linear birational transformations on the entries of a q q matrix this mapping originates from the analysis of birational transformations obtained from very simple algebraic calculations namely taking the inverse of q q matrices and permuting some of the entries of these matrices it has been seen to yield weak chaos and integrability we have found new integrable cases of this cremona transformation corresponding to the values of 0 when e 12 13 1 besides the already known values e 0 and e 1 and also arbitrary when e 0 for these cases one has a foliation of the parameter space in elliptic curves we give the equations of these elliptic curves based on this very example we show how one can find these integrability cases of the cremona transformation and actually integrate it using a method based on the systematic study of the finite order conditions of the cremona transformation the method is shown to be efficient and straightforward the various integrability cases are revisited using many different representations of this very mapping birational transformations recursion in one variable

real topological entropy versus metric entropy for birational measure preserving transformations abstract we consider a family of birational measure preserving transformations of two complex variables depending on one parameter for which simple rational expressions for the dynamical zeta function have been conjectured together with an equality between the topological entropy and the logarithm of the arnold complexity divided by the number of iterations similar results have been obtained for the adaptation of these two concepts to dynamical systems of real variables yielding to introduce a real topological entropy and a real arnold complexity we try to compare here the kolmogorov sinai metric entropy and this real arnold complexity or real topological entropy on this particular example of a one parameter dependent birational transformation of two variables more precisely we analyze using an infinite precision calculation the lyapunov characteristic exponents for various values of the parameter of the birational transformation in order to compare these results with the ones for the real arnold complexity we find a quite surprising result for this very birational example and in fact for a large set of birational measure preserving mappings generated by involutions the lyapunov characteristic exponents seem to be equal to zero or at least extremely small for all the orbits we have considered and for all values of the parameter birational measure preserving transformations generated by involutions could thus allow to better understand the difference between the topological description and the probabilistic description of discrete dynamical systems many birational measure preserving transformations generated by involutions seem to provide examples of discrete dynamical systems which can be topologically chaotic while they are metrically almost quasi periodic heuristically this can be understood as a consequence of the fact that their orbits seem to form some kind of transcendental foliation of the two dimensional space of variables

square lattice ising model susceptibility series expansion method and differential equation for χ 3 in a previous paper 2004 j phys a math gen 37 9651 68 we have given the fuchsian linear differential equation satisfied by 3 the three particle contribution to the susceptibility of the isotropic square lattice ising model this paper gives the details of the calculations with some useful tricks and tools allowing one to obtain long series in polynomial time the method is based on series expansion in the variables that appear in the n 1 dimensional integrals representing the n particle contribution to the isotropic square lattice ising model susceptibility the integration rules are straightforward due to remarkable formulae we derived for these variables we obtain without any numerical approximation 3 as a fully integrated series in the variable w s 2 1 s2 where s sh 2k with k j kt the conventional ising model coupling constant we also give some perspectives and comments on these results

generalized bsdes and nonlinear neumann boundary value problems we study a new class of backward stochastic differential equations which involves the integral with respect to a continuous increasing process this allows us to give a probabilistic formula for solutions of semilinear partial differential equations with neumann boundary condition where the boundary condition itself is nonlinear we consider both parabolic and elliptic equations