sequentiality monadic second order logic and tree automata abstract given a term rewriting system r and a normalizable term t a redex is needed if in any reduction sequence of t to a normal form this redex will be contracted roughly r is sequential if there is an optimal reduction strategy in which only needed redexes are contracted more generally g huet and j j levy have defined the sequentiality of a predicate p on partially evaluated terms 1991 computational logic essays in honor of alan robinson mit press cambridge ma pp 415 443 we show here that the sequentiality of p is definable in sks the monadic second order logic with k successors provided p is definable in sks we derive several known and new consequences of this remark 1 strong sequentiality as defined by huet and levy of a left linear possibly overlapping rewrite system is decidable 2 nv sequentiality as defined in m oyamaguchi 1993 siam j comput 19 424 437 is decidable even in the case of overlapping rewrite systems 3 sequentiality of any linear shallow rewrite system is decidable then we describe a direct construction of a tree automaton recognizing the set of terms that do have needed redexes which again yields immediate consequences 1 strong sequentiality of possibly overlapping linear rewrite systems is decidable in exptime 2 for strongly sequential rewrite systems needed redexes can be read directly on the automaton

sl 3 r as the group of symmetry transformations for all one dimensional linear systems ii realizations of the lie algebra the two dimensional space time realizations of the lie algebra of sl 3 r are obtained when the group acts as the maximal point symmetry group of any given one dimensional newtonian linear system it is shown that these realizations are isomorphic with the realization of the lie algebra of the projective group in the plane next an active point of view is introduced and sl 3 r is interpreted as a group of mappings that transform one admissible world line of the system into another thus a new mechanical realization of the sl 3 r algebra comes to the fore some miscellaneous examples are included

on connections of the lienard equation with some equations of painleve gambier type abstract the lienard equation is used in various applications therefore constructing general analytical solutions of this equation is an important problem here we study connections between the lienard equation and some equations from the painleve gambier classification we show that with the help of such connections one can construct general analytical solutions of the lienard equation s subfamilies in particular we find three new integrable families of the lienard equation we also propose and discuss an approach for finding one parameter families of closed form analytical solutions of the lienard equation

the covariance matrix of the potts model a random cluster analysis we consider the covariance matrix g mm q2 of thed dimensionalq states potts model rewriting it in the random cluster representation of fortuin and kasteleyn in any of theq ordered phases we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances thereby attributing algebraic significance to these stochastic geometric quantities we also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers for dimensiond 2 we show that this correlation length and the correlation length of the two point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two for systems with first order transitions this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self dual point o for systems with second order transitions this relation implies the equality of the correlation length exponents from above and below threshold as well as an amplitude ratio of two in the course of proving the above results we establish several properties of independent interest including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions and left continuity of the two dimensional free boundary condition percolation probability at o we also introduce dlr equations for the random cluster model and use them to establish ergodicity of the free measure in order to prove these results we introduce a new class of events which we call decoupling events and two inequalities for these events the first is similar to the fkg inequality but holds for events which are neither increasing nor decreasing the second is similar to the van den berg kesten inequality in standard percolation both inequalities hold for an arbitrary fkg measure

nonstandard conserved hamiltonian structures in dissipative damped systems nonlinear generalizations of damped harmonic oscillator in this paper we point out the existence of a remarkable nonlocal transformation between the damped harmonic oscillator and a modified emden type nonlinear oscillator equation with linear forcing x xx x3 x 0 which preserves the form of the time independent integral conservative hamiltonian and the equation of motion generalizing this transformation we prove the existence of nonstandard conservative hamiltonian structure for a general class of damped nonlinear oscillators including lienard type systems further using the above hamiltonian structure for a specific example namely the generalized modified emden equation x xqx x2q 1 0 where and q are arbitrary parameters the general solution is obtained through appropriate canonical transformations we also present the conservative hamiltonian structure of the damped mathews lakshmanan oscillator equation the associated lagrangian description for all the above systems is also briefly discussed

analysis and solution of a nonlinear second order differential equation through rescaling and through a dynamical point of view the solutions of the equation y yy y3 0 where is a free parameter are investigated for 1 9 the equation is linearizable through an eight parameter symmetry group and is completely integrable for 1 9 only two symmetries subsist but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of and according to the initial conditions

the branching random walk and contact process on galton watson and nonhomogeneous trees we show that the branching random walk on a galton watson tree may have one or two phase transitions depending on the relative sizes of the mean degree and the maximum degree we show that there are some galton watson trees on which the branching random walk has one phase transition while the contact process has two this contradicts a conjecture of madras and schinazi we show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree contradicting a conjecture of pemantle

infinitesimal symmetry transformations of some one dimensional linear systems the converse problem of similarity analysis is solved in general for the infinitesimal symmetry transformations of any given inhomogeneous ordinary differential equation of the second order x f2 t x f1 t x f0 t the completely general associated lie algebra is obtained for equations of this kind which structure constants depend only on the chosen set of initial values f0 0 f1 0 f2 0 and f2 0 the infinitesimal elements of the dynamical group of a newtonian one dimensional linear system are also discussed and some miscellaneous examples are given

a nonlocal connection between certain linear and nonlinear ordinary differential equations oscillators we explore a nonlocal connection between certain linear and nonlinear ordinary differential equations odes representing physically important oscillator systems and identify a class of integrable nonlinear odes of any order we also devise a method to derive explicit general solutions of the nonlinear odes interestingly many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

lax representation and quadratic first integrals for a family of non autonomous second order differential equations abstract we consider a family of non autonomous second order differential equations which generalizes the lienard equation we explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic with the respect to the first derivative first integrals we show that these conditions are equivalent to the conditions for equations in the family under consideration to possess lax representations this provides a connection between the existence of a quadratic first integral and a lax representation for the studied dissipative differential equations which may be considered as an analogue to the theorem that connects lax integrability and arnold liouville integrability of hamiltonian systems we illustrate our results by several examples of dissipative equations including generalizations of the van der pol and duffing equations each of which have both a quadratic first integral and a lax representation

infinite clusters in percolation models the qualitative nature of infinite clusters in percolation models is investigated the results which apply to both independent and correlated percolation in any dimension concern the number and density of infinite clusters the size of their external surface the value of their total surface to volume ratio and the fluctuations in their density in particular it is shown thatn0 the number of distinct infinite clusters is either 0 1 or and the casen0 which might occur in sufficiently high dimension is analyzed

ising models on hyperbolic graphs we consider ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph v f where v is the number of neighbors of each vertex and f is the number of sides of each face let tc be the critical temperature and t c sup t tc f 2 where f is the free boundary condition b c gibbs state is the plus b c gibbs state and is the minus b c gibbs state we prove that if the hyperbolic graph is self dual i e v f or if v is sufficiently large how large depends on f e g v 35 suffices for any f 3 and v 17 suffices for any f 17 then 0 t c tc in contrast with that t c tc for ising models on the hypercubic lattice zd with d 2 a result due to lebowitz 22 while whenever t t c f 2 the last result is an improvement in comparison with the analogous statement in refs 28 and 33 in which it was only proved that f 2 when t t c and it remains to show in both papers that f 2 whenever t t c therefore t c and tc divide 0 into three intervals 0 t c t c tc and tc in which but f 2 and f 2 and respectively

on the general solution for the modified emden type equation in this paper we demonstrate that the modified equation of emden type mee is integrable either explicitly or by quadrature for any value of and we also prove that the mee possesses appropriate time independent hamiltonian functions for the full range of parameters and in addition we show that the mee is intimately connected with two well known nonlinear models namely the equation of force free duffing type oscillator and the two dimensional lotka volterra equation and thus the complete integrability of the latter two models can also be understood in terms of the mee

uniqueness of the infinite component in a random graph with applications to percolation and spin glasses we extend the theorem of burton and keane on uniqueness of the infinite component in dependent percolation to cover random graphs on d or d with long range edges we also study a short range percolation model related to nearest neighbor spin glasses on d or on a slab d 0 k and prove both that percolation occurs and that the infinite component is unique forv 2 0 1 or larger

indistinguishability of percolation clusters we show that when percolation produces infinitely many infinite clusters on a cayley graph one cannot distinguish the clusters from each other by any invariantly defined property this implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity a k a long range order we then derive applications concerning uniqueness in kazhdan groups and in wreath products and inequalities for p u

random walks capacity and percolation on trees a collection of several different probabilistic processes involving trees is shown to have an unexpected unity this makes possible a fruitful interplay of these probabilistic processes the processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well our work has five specific aims first an exact correspondence between random walks and percolation on trees is proved extending and sharpening previous work of the author this is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes second we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation this capacitary formulation extends and sharpens work of fan on random interval coverings third we show how this formulation also applies to generalize work of evans on random labelling of trees fourth the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a s in particular we resolve a conjecture of griffeath on the necessity of the nash williams criterion fifth for this last purpose we establish several new basic results on branching processes in varying environments

sl 3 r realisations and the damped harmonic oscillator the point and contact lie groups for the damped harmonic oscillator are found and several realisations of the sl 3 r lie group are analysed the non critical and critical cases are studied separately and a sort of limiting procedure is defined which leaves the lie algebra structure unchanged some physical applications of these results are also pointed out

phase diagram of the ising model on a cayley tree in the presence of competing interactions and magnetic field we study the phase diagram for the ising model on a cayley tree with competing nearest neighbor interactionsj1 and next nearest neighbor interactionsj2 andj3 in the presence of an external magnetic field to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular cases previous works by vannimenus and by inawashiroet al at vanishing temperature the phase diagram is fully determined for all values and signs ofj2 j1 andj3 j2 in particular we verify that values ofj3 j2 high enough favor the paramagnetic phase at finite temperatures several interesting features evolution of reentrances separation of the modulated region into two disconnected pieces etc are exhibited for typical values ofj2 j1 andj3 j2

difference schemes with point symmetries and their numerical tests symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented they have the same three or four dimensional symmetry groups as the original differential equations the new difference schemes are tested as numerical methods the obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost for an example involving a solution with a singularity in the integration region the symmetry preserving scheme contrary to standard ones provides solutions valid beyond the singular point

exact solution of an ising model with competing interactions on a cayley tree the exact solution of an ising model with competing restricted interactions on the cayley tree and in the absence of an external field is presented a critical curve is defined where it is possible to get phase transitions above it and a single gibbs state is obtained elsewhere

on gibbs measures of models with competing ternary and binary interactions and corresponding von neumann algebras in the present paper the ising model with competing binary j and binary j1 interactions with spin values 1 on a cayley tree of order 2 is considered the structure of gibbs measures for the model is studied we completely describe the set of all periodic gibbs easures for the model with respect to any normal subgroup of finite index of a group representation of the cayley tree types of von neumann algebras generated by gns representation associated with diagonal states corresponding to the translation invariant gibbs measures are determined it is proved that the factors associated with minimal and maximal gibbs states are isomorphic and if they are of type iii then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively some concrete examples of factors are given too

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

the painleve test hidden symmetries and the equation y yy ky3 0 for general values of the parameter k the equation y yy ky3 0 can be reduced to quadrature via a lie algebraic approach either direct or through hidden symmetries for specific values of k mostly in 1 9 1 8 the solution can be expressed in parametric form for these values of k the equation passes the weak painleve test for some other values of k the equation passes the painleve test but the solution cannot in general be expressed parametrically

lie and noether symmetry groups of nonlinear equations it is found that the nonlinear ordinary differential equation rho omega 2 t rho 1 rho 3 has a three parameter lie group of symmetries which are also noether symmetries the invariants associated with the group are calculated the authors discuss a new way of generating n parameter lie symmetry groups which are associated with n parameter nonlinear differential equations

on the singularity analysis of ordinary differential equations invariant under time translation and rescaling the pt is applied to the general second and third order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability the effect of the two symmetries on the compatibility conditions is determined and we show that generally these conditions are automatically a consequence of the resonance condition use is made of truncated laurent series both in ascending and descending powers as an example the case of the generalized chazy equation is presented

the contact process on trees the contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter is varied for small values of a single infection eventually dies out for larger the infection lives forever with positive probability but eventually leaves any finite set the survival probability is a continuous function of and the proof of this is much easier than it is for the contact process on d dimensional integer lattices for still larger the infection converges in distribution to a nontrivial invariant measure for an n ary tree with n large the first of these transitions occurs when 1 n and the second occurs when 1 2 n e n nonhomogeneous trees whose vertices have degrees varying between 1 and n behave essentially as homogeneous n ary trees provided that vertices of degree n are not too rare in particular letting n go to galton watson trees whose vertices have degree n with probability that does not decrease exponentially with n may have both phase transitions occur together at 0 the nature of the second phase transition is not yet clear and several problems are mentioned in this regard

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

ising models on the lobachevsky plane we consider the ising model on a lattice which is the orbit of a discrete cocompact group acting on the hyperbolic plane for large values of the inverse temperature we construct an uncountable number of mutually singular gibbs states

nv sequentiality a decidable condition for call by need computations in term rewriting systems in 1979 huet and levy introduced the class of sequential term rewriting systems in which call by need computations are possible without look ahead and defined the subclass called strongly sequential systems for which needed redexes in a given term are effectively found chapter in computational logic essays in honor of alan robinson j l lassez and g plotkin eds mit press cambridge ma 1991 this paper introduces a larger subclass that is a natural extension of strong sequentiality and is based on the analysis of both the left hand sides and part of the right hand sides i e the nonvariable parts of systems whereas strong sequentiality is based on the analysis of left hand sides alone this new sequentiality is called nv sequentiality it is shown that i the class of nv sequential systems properly includes the class of strongly sequential systems ii there exists an algorithm for finding needed redexes for a given term when a system is nv sequential and iii it is decidable whether a

nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order in this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations odes of arbitrary order related to linear odes through nonlocal relations the procedure makes use of the lie point symmetries of the linear odes and the nonlocal connection to deduce the nonlocal symmetries of the corresponding nonlinear odes using these nonlocal symmetries we obtain reduction transformations and reduced equations to specific examples we find that the reduced equations can be explicitly integrated to deduce the general solutions for these cases we also extend this procedure to coupled higher order nonlinear odes with specific reference to second order nonlinear odes

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

monotonicity of uniqueness for percolation on cayley graphs all infinite clusters are born simultaneously consider site or bond percolation with retention parameter p on an infinite cayley graph in response to questions raised by grimmett and newman 1990 and benjamini and schramm 1996 we show that the property of having almost surely a unique infinite open cluster is increasing in p moreover in the standard coupling of the percolation models for all parameters a s for all p2 p1 pc each infinite p2 cluster contains an infinite p1 cluster this yields an extension of alexander s 1995 simultaneous uniqueness theorem as a corollary we obtain that the probability v p that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p pc all our results extend to quasi transitive infinite graphs with a unimodular automorphism group

the ising model on diluted graphs and strong amenability say that a graph has persistent transition if the ising model on the graph can exhibit a phase transition nonuniqueness of gibbs measures in the presence of a nonzero external field we show that for nonamenable graphs for bernoulli percolation with p close to 1 all the infinite clusters have persistent transition on the other hand we show that for transitive amenable graphs the infinite clusters for any stationary percolation do not have persistent transition this extends a result of georgii for the cubic lattice a geometric consequence of this latter fact is that the infinite clusters are strongly amenable i e their anchored cheeger constant is 0 finally we show that the critical temperature for the ising model with no external field on the infinite clusters of bernoulli percolation with parameter p on an arbitrary bounded degree graph is a continuous function of p

surface structure of random aggregates on the cayley tree growth through ballistic aggregation and biased diffusion limited aggregation is investigated on the cayley tree for a general branching ratio it is shown that the surface width of a ballistic aggregate remains finite of order one as the cluster mass goes to infinity dla with an attractive bias is treated approximately for non zero bias strength the surface width is asymptotically finite in the limit of isotropic diffusion zero bias a roughening transition occurs which can be described in terms of a single diverging length scale

sl 3 r as the group of symmetry transformations for all one dimensional linear systems the converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order x f2 t x f1 t x f0 t the eight parameter realizations of the symmetry group are obtained in the form f 1p2 f where f stands for transformations of t x that depend exclusively on the fundamental solutions of the equation and where p2 is an arbitrary projective transformation in the plane thus it is shown that the full point symmetry group corresponds to sl 3 r indeed without recourse to the lie algebra also a technique is obtained for calculating the finite point symmetry realization of sl 3 r for any given one dimensional linear system some miscellaneous examples are given

decidable call by need computations in term rewriting the theorem of huet and levy stating that for orthogonal rewrite systems i every reducible term contains a needed redex and ii repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form forms the basis of all results on optimal normalizing strategies for orthogonal rewrite systems however needed redexes are not computable in general in the paper we show how the use of approximations and elementary tree automata techniques allows one to obtain decidable conditions in a simple and elegant way surprisingly by avoiding complicated concepts like index and sequentiality we are able to cover much larger classes of rewrite systems we also study modularity aspects of the classes in our hierarchy it turns out that none of the classes is preserved under signature extension by imposing various conditions we recover the preservation under signature extension by imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructor sharing combinations

nonmonotonic behavior in hard core and widom rowlinson models we give two examples of nonmonotonic behavior in symmetric systems exhibiting more than one critical point at which spontaneous symmetry breaking appears or disappears the two systems are the hard core model and the widom rowlinson model and both examples take place on a variation of the cayley tree bethe lattice devised by schonmann and tanaka we obtain similar though less constructive examples of non monotonicity via certain local modifications of any graph e g the square lattice which is known to have a critical point for either model en route we prove that the widom rowlinson model does behave monotonically on the ordinary cayley tree and we obtain an explicit description of its critical behavior there we conclude with some results about monotonicity of the phase transition phenomenon relative to graph structure

context sensitive rewriting strategies contex sensitive rewriting is a simple rewriting restriction which is formalized by imposing fixed restrictions on replacements such a restriction is given on a purely syntactic basis it is given on the arguments of symbols of the signature and inductively extended to arbitrary positions of terms built from those symbols the termination behavior is not only preserved but usually improved and several methods have been developed to formally prove it in this paper we investigate the definition properties and use of context sensitive rewriting strategies i e particular fixed sequences of context sensitive rewriting steps we study how to define them in order to obtain efficient computations and to ensure that context sensitive computations terminate whenever possible we give conditions enabling the use of these strategies for root normalization normalization and infinitary normalization we show that this theory is suitable for formalizing the definition and analysis of real computational strategies which are used in programming languages such as obj or elan

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

unusual lienard type nonlinear oscillator a lienard type nonlinear oscillator of the form x kxx k2 9 x3 lambda1x 0 which may also be considered as a generalized emden type equation is shown to possess unusual nonlinear dynamical properties it is shown to admit explicit nonisolated periodic orbits of conservative hamiltonian type for lambda1 0 these periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator this is completely contrary to the standard characteristic property of nonlinear oscillators interestingly the system though appears deceptively a dissipative type for lambda1 or 0 does admit a conserved hamiltonian description where the characteristic decay time is also independent of the amplitude the results also show that the criterion for conservative hamiltonian system in terms of divergence of flow function needs to be generalized

lie algebras of vector fields in the real plane finite dimensional real analytic lie algebras of vector fields on r 2 are completely classified up to changes of local coordinates

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

a new proof that for the contact process on homogeneous trees local survival implies complete convergence we provide a new proof substantially simpler than zhang s original one that for the contact process on homogeneous trees local survival implies complete convergence

an ising model with three competing interactions on a cayley tree in this paper we consider an ising model with three competing restricted interactions on the cayley tree j2 j3 the translation invariant and periodic gibbs measures for these models are investigated and the problem of the phase transition in these classes is solved

stochastic models of interacting systems interacting particle systems is by now a mature area of probability theory but one that is still very active we begin this paper by explaining how models from this area arise in fields such as physics and biology we turn then to a discussion of both older and more recent results about them concentrating on contact processes voter models and exclusion processes these processes are among the most studied in the field and have the virtue of relative simplicity in their description which permits us to address the fundamental issues about their behavior without dealing with the extra complications that models from specific areas of application would require

phase transitions for quantum markov chains associated with ising type models on a cayley tree the main aim of the present paper is to prove the existence of a phase transition in quantum markov chain qmc scheme for the ising type models on a cayley tree note that this kind of models do not have one dimensional analogous i e the considered model persists only on trees in this paper we provide a more general construction of forward qmc in that construction a qmc is defined as a weak limit of finite volume states with boundary conditions i e qmc depends on the boundary conditions our main result states the existence of a phase transition for the ising model with competing interactions on a cayley tree of order two by the phase transition we mean the existence of two distinct qmc which are not quasi equivalent and their supports do not overlap we also study some algebraic property of the disordered phase of the model which is a new phenomena even in a classical setting

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

intermediate phase for the contact process on a tree we show that in between the two critical values for the contact process on trees there are infinitely many extremal nontranslation invariant stationary distributions this conclusion is reminiscent of results of grimmett and newman for percolation on the product of a tree with the integers where there is an intermediate phase with infinitely many infinite clusters

tree indexed random walks on groups and first passage percolation suppose that i i d random variables are attached to the edges of an infinite tree when the tree is large enough the partial sumss along some of its infinite paths will exhibit behavior atypical for an ordinary random walk this principle has appeared in works on branching random walks first passage percolation and rwre on trees we establish further quantitative versions of this principle which are applicable in these settings in particular different notions of speed for such a tree indexed walk correspond to different dimension notions for trees finally if the labeling variables take values in a group then properties of the group e g polynomial growth or a nontrivial poisson boundary are reflected in the sample path behavior of the resulting tree indexed walk

new monte carlo method for the self avoiding walk we introduce a new monte carlo algorithm for the self avoiding walk saw and show that it is particularly efficient in the critical region long chains we also introduce new and more efficient statistical techniques we employ these methods to extract numerical estimates for the critical parameters of the saw on the square lattice we find 2 63820 0 00004 0 00030 1 352 0 006 0 025 v 0 7590 0 0062 0 0042 where the first error bar represents systematic error due to corrections to scaling subjective 95 confidence limits and the second bar represents statistical error classical 95 confidence limits these results are based on saws of average length 166 using 340 hours cpu time on a cdc cyber 170 730 we compare our results to previous work and indicate some directions for future research

continuity of percolation probability on hyperbolic graphs let t be a ibrwarding tree of degree k where each vertex other than the origin has k children and one parent and the origin has k children but no parent k 2 define g to be the graph obtained by adding to t nearest neighbor bonds connecting the vertices which are in the same generation g is regarded as a discretization of the hyperbolic plane h in the same sense that z a is a discretization of r a independent percolation on g has been proved to have multiple phase transitions we prove that the percolation probability l p is continuous on 0 1 as a function ofp

a quantum exactly solvable nonlinear oscillator related to the isotonic oscillator a nonpolynomial one dimensional quantum potential representing an oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator harmonic oscillator with a centripetal barrier is studied first the general case that depends on a parameter a is considered and then a particular case is studied with great detail it is proven that it is schr dinger solvable and then the wavefunctions n and the energies en of the bound states are explicitly obtained finally it is proven that the solutions determine a family of orthogonal polynomials related to the hermite polynomials and such that i every is a linear combination of three hermite polynomials and ii they are orthogonal with respect to a new measure obtained by modifying the classic hermite measure

hidden symmetries and linearization of the modified painleve ince equation the linearization and hidden symmetries of the modified painleve ince equation y yy y3 0 where and are constants are presented the linearization of this equation by a nonlocal transformation yields a damped stable or growing unstable harmonic oscillator equation for 0 hidden symmetries are analyzed by transforming the modified painleve ince equation to a third order ordinary differential equation ode which in general is invariant under a three parameter group by a riccati transformation a type i hidden symmetry is found of a second order ode found from the third order ode where a symmetry is lost in the reduction of order by the non normal subgroup invariants a type ii hidden symmetry occurs in the third order ode because the symmetries of a second order ode reduced from the third order ode by another set of normal subgroup invariants are increased

noether s theorem and lie symmetries for time dependent hamilton lagrange systems noether and lie symmetry analyses based on point transformations that depend on time and spatial coordinates will be reviewed for a general class of time dependent hamiltonian systems the resulting symmetries are expressed in the form of generators whose time dependent coefficients follow as solutions of sets of ordinary differential auxiliary equations the interrelation between the noether and lie sets of auxiliary equations will be elucidated the auxiliary equations of the noether approach will be shown to admit invariants for a much broader class of potentials compared to earlier studies as an example we work out the noether and lie symmetries for the time dependent kepler system the runge lenz vector of the time independent kepler system will be shown to emerge as a noether invariant if we adequately interpret the pertaining auxiliary equation furthermore additional nonlocal invariants and symmetries of the kepler system will be isolated by identifying further solutions of the auxiliary equations that depend on the explicitly known solution path of the equations of motion showing that the invariants remain unchanged under the action of different symmetry operators we demonstrate that a unique correlation between a symmetry transformation and an invariant does not exist

multiple transition points for the contact process on the binary tree the contact process on z d is known to have only two fundamental types of behavior survival and extinction recently pemantle discovered that the phase structure for the contact process on a tree can be more complex there are three possible types of behavior strong survival weak survival and extinction he proved that all three occur on homogeneous trees in which each vertex has d 1 neighbors provided that d 3 but he left open the case d 2 since d 1 corresponds to z 1 in which weak survival does not occur d 2 is the boundary case in this paper we complete this picture by showing that weak survival does occur on the binary tree for appropriate parameter values in doing so we extend and develop techniques for obtaining upper and lower bounds for the critical values associated with strong and weak survival of the contact process on more general graphs

a sequential reduction strategy kennaway proved the remarkable result that every almost orthogonal term rewriting system admits a computable sequential normalizing reduction strategy in this paper we present a computable sequential reduction strategy similar in scope but simpler and more general our strategy can be thought of as an outerrnost fair like strategy that is allowed to be unfair to some redex of a term when contracting the redex is useless for the normalization of the term unlike the strategy of kennaway our strategy does not rely on syntactic restrictions that imply confluence on the contrary it can easily be applied to any term rewriting system and we show that the class of term rewriting systems for which our strategy is normalizing properly includes all almost orthogonal systems our strategy is more versatile in case of almost orthogonal term rewriting systems it can be used to detect certain cases of non termination our normalization proof is more accessible than kennaway s we also show that our sequential strategy sometimes succeeds where the parallel outermost strategy fails

random walks on the lamplighter group kaimanovich and vershik described certain finitely generated groups of exponential growth such that simple random walk on their cayley graph escapes from the identity at a sublinear rate or equivalently all bounded harmonic functions on the cayley graph are constant here we focus on a key example called g 1 by kaimanovich and vershik and show that inward biased random walks on g 1 move outward faster than simple random walk indeed they escape from the identity at a linear rate provided that the bias parameter is smaller than the growth rate of g 1 these walks can be viewed as random walks interacting with a dynamical environment on mathbb z the proof uses potential theory to analyze a stationary environment as seen from the moving particle

integrability and symmetries for the helmholtz oscillator with friction this paper deals with the helmholtz oscillator which is a simple nonlinear oscillator whose equation presents a quadratic nonlinearity and the possibility of escape when a periodic external force is introduced the width of the stochastic layer which is a region around the separatrix where orbits may exhibit transient chaos is calculated in the absence of friction and external force it is well known that analytical solutions exist since it is completely integrable when only friction is included there is no analytical solution for all parameter values however by means of the lie theory for differential equations we find a relation between parameters for which the oscillator is integrable this is related to the fact that the system possesses a symmetry group and the corresponding symmetries are computed finally the analytical explicit solutions are shown and related to the basins of attraction

a simple and unified approach to identify integrable nonlinear oscillators and systems in this paper we consider a generalized second order nonlinear ordinary differential equation ode of the form x k1xq k2 x k3x2q 1 k4xq 1 1x 0 where ki s i 1 2 3 4 1 and q are arbitrary parameters which includes several physically important nonlinear oscillators such as the simple harmonic oscillator anharmonic oscillator force free helmholtz oscillator force free duffing and duffing van der pol oscillators modified emden type equation and its hierarchy generalized duffing van der pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation we identify several new integrable cases for arbitrary value of the exponent q q r the q 1 and q 2 cases are analyzed in detail and the results are generalized to arbitrary q our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature t

markov fields on branching planes we investigate ising models indexed by the sites of a branching plane mathbb t which is the product of a regular tree mathbb t and the line there are three parameter regimes corresponding to 1 a unique gibbs distribution 2 nonunique gibbs distributions with treelike structure the free boundary condition field is not a mixture of the plus and minus b c fields 3 nonunique gibbs distributions with planelike structure the free b c field is a mixture of the plus and minus b c fields

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

phase transition in reinforced random walk and rwre on trees a random walk on an infinite tree is given a particular kind of positive feedback so edges already traversed are more likely to be traversed in the future using exchangeability theory the process is shown to be equivalent to a random walk in a random environment rwre that is to say a mixture of markov chains criteria are given to determine whether a rwre is transient or recurrent these criteria apply to show that the reinforced random walk can vary from transient to recurrent depending on the value of an adjustable parameter measuring the strength of the feedback the value of the parameter at the phase transition is calculated

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems