linearizable mappings and the low growth criterion we examine a family of discrete second order systems which are integrable through reduction to a linear system these systems were previously identified using the singularity confinement criterion here we analyse them using the more stringent criterion of nonexponential growth of the degrees of the iterates we show that the linearizable mappings are characterized by a very special degree growth the ones linearizable by reduction to projective systems exhibit zero growth i e they behave like linear systems while the remaining ones derivatives of riccati and gambier mapping lead to linear growth this feature may well serve as a detector of integrability through linearization

the last remake of the gambier mapping abstract we present a special discrete analogue of the gambier coupled riccati system where the coupling is introduced in an additive way we show that there exist choices of the parameters for which the singularities of the mapping are confined computing the continuous limit of the mapping we show that the resulting differential equation never possesses the painleve property we present an explanation of this apparent paradox

symmetry relations in exactly soluble models symmetry relations such as the star triangle or the inverse relation are very useful in determining the partition function of two dimensional exactly soluble models a common construction of the three dimensional equivalents of these symmetry relations is presented they are used to derive in a geometric way applying simultaneously to different kinds of spin models the consequent global properties i e the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function the usefulness of the inverse relation is illustrated by an application to the three dimensional ising model

integrability from 2d n 2 2 dualities we study integrable models in the context of the recently discovered gauge ybe correspondence where the yang baxter equation is promoted to a duality between two su persymmetric gauge theories we study flavored elliptic genus of 2d n 2 2 quiver gauge theories which theories are defined from statistical lattices regarded as quiver diagrams our r matrices are written in terms of theta functions and simplifies considerably when the gauge groups at the quiver nodes are abelian we also discuss the modularity properties of the r matrix reduction of 2d index to 1d witten index and string theory realizations of our theories

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

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 ultimate estimate of the upper norm bound for the summation of operators abstract let a and b be bounded linear operators acting on a hilbert space h it is shown that the triangular inequality serves as the ultimate estimate of the upper norm bound for the sum of two operators in the sense that sup u au v bv u and v are unitaries min a i b i c consequences of the result related to spectral sets the von neumann inequality and normal dilations are discussed furthermore it is shown that the above equality can be used to characterize those unitarily invariant norms that are multiples of the operator norm in the finite dimensional case

functional relations and nested bethe ansatz 3 chiral potts model at q 2 1 we obtain the functional relations for the eigenvalues of the transfer matrix of the sl 3 chiral potts model for for the homogeneous model in both directions a solution of these functional relations can be written in terms of roots of bethe ansatz like equations in addition a direct nested bethe ansatz has also been developed for this case

commuting quantum traces the case of reflection algebras we formulate a systematic construction of commuting quantum traces for reflection algebras this is achieved by introducing two dual sets of generalized reflection equations with associated consistent fusion procedures products of their respective solutions yield commuting quantum traces

two state spectral free solutions of the frenkel moore simplex equation whilst many solutions have been found for the quantum yang baxter equation qybe there are fewer known solutions available for its higher dimensional generalizations zamolodchikov s tetrahedron equation zte and frenkel and moore s simplex equation fme in this paper we present families of solutions to the fme which may help us to understand more about higher dimensional generalization of the qybe

dilogarithm identities in conformal field theory and group homology recently rogers dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations one of the connecting threads is an identity of richmond szekeres that appeared in the computation of central charges in conformal field theory we show that the richmond szekeres identity and its extension by kirillov reshetikhin equivalent to an identity found earlier by lewin can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all 2 2 real matrices viewed as adiscrete group this connection allows us to clarify a few of the assertions and conjectures stated in the work of nahm recknagel terhoven concerning the role of algebraic k theory and thurston s program on hyperbolic 3 manifolds specifically it is not related to hyperbolic 3 manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all 2 2 real matrices viewed as a topological group this also resolves the weaker version of the conjecture as formulated by kirillov we end with a summary of a number of open conjectures on the mathematical side

towards an exhaustive classification of the star triangle relation ii for pt i see ibid vol 17 p 1251 1984 the authors continue their study of the star triangle relation for two component spin models with the help of the inversion relation the conclusion is that the most general interaction round a face models 16 non zero parameters that satisfy the star triangle relations are mainly baxter 1972 and fan and wu 1970 models in this way they recover krichever s result 1981 for the classification of the star triangle relations for vertex models they sketch a generalisation of this approach to models with excluded configurations to staggered models and to three dimensional models tetrahedron relations

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

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

critical correlations in a z invariant inhomogeneous ising model abstract we study the critical spin spin correlation function in the z invariant inhomogeneous ising model which includes the rectangular triangular honeycomb and checkerboard lattices as special cases using quadratic difference equations we derive various explicit representations for the correlation function confirming and simplifying also a miraculous result of baxter

new solutions of reflection equation derived from type b bmw algebras we use b type knot theory to find new solutions of sklyanin s reflection equation in a systematic way this generalizes the well known baxterization of birman wenzl algebras and should describe integrable systems which are restricted to a half plane

on zamolodchikov s solution of the tetrahedron equations the tetrahedron equations arise in field theory as the condition for thes matrix in 2 1 dimensions to be factorizable and in statistical mechanics as the condition that the transfer matrices of three dimensional models commute zamolodchikov has proposed what appear from numerical evidence and special cases to be non trivial particular solutions of these equations but has not fully verified them here it is proved that they are indeed solutions

an integrable 3d lattice model with positive boltzmann weights in this paper we construct a three dimensional 3d solvable lattice model with non negative boltzmann weights the spin variables in the model are assigned to edges of the 3d cubic lattice and run over an infinite number of discrete states the boltzmann weights satisfy the tetrahedron equation which is a 3d generalization of the yang baxter equation the weights depend on a free parameter 0 q 1 and three continuous field variables the layer to layer transfer matrices of the model form a two parameter commutative family this is the first example of a non trivial solvable 3d lattice model with non negative boltzmann weights

baxter equations and the zamolodchikov model abstract a summary is given of the significance of the star triangle or yang baxter equations in the solution of two dimensional statistical mechanical models the extension to the three dimensional tetrahedron relations is quoted and it is shown how this can be used to obtain the partition function of the zamolodchikov model for an n lattice the n result differs from that recently obtained by bazhanov and stroganov

practical method for determining the minimum embedding dimension of a scalar time series abstract a practical method is proposed to determine the minimum embedding dimension from a scalar time series it has the following advantages 1 does not contain any subjective parameters except for the time delay for the embedding 2 does not strongly depend on how many data points are available 3 can clearly distinguish deterministic signals from stochastic signals 4 works well for time series from high dimensional attractors 5 is computationally efficient several time series are tested to show the above advantages of the method

diagonalization of the transfer matrix of a nonintersecting string model abstract in this paper we obtain the eigenvectors of the transfer matrix of a nonintersecting string model using the bethe ansatz the result is in agreement with baxter s conjecture of weak equivalence of the present model with the six vertex model

quantum 2 1 evolution model a quantum evolution model in 2 1 discrete spacetime connected with a 3d fundamental map is investigated map is derived as a map providing a zero curvature of a 2d linear lattice system called the current system in a special case of the local weyl algebra for dynamical variables the map appears to be canonical and it corresponds to the known operator valued matrix the current system is a type of the linear problem for the 2 1 evolution model a generating function for the integrals of motion for the evolution is derived with the help of the current system thus the complete integrability in 3d is proved directly

finding lie groups that reduce the order of discrete dynamical systems discrete dynamical systems of the form are considered where is an n component vector equations x f x define a mapping f from an n dimensional projective space into itself each component of f is a rational function i e a ratio of polynomials in n dynamical variables maeda showed that when f commutes with each transformation of a lie group a reduction in the order of the dynamical system results given a discrete dynamical system the difficulty is to find its continuous symmetries we present a way of using f invariant sets to find these symmetries the approach taken is to arrange groups in order of increasing complication and to characterize the set of dynamical systems admitting each group criteria are given for recognizing and reducing the order of systems admitting subgroups of the projective general linear group in n variables pgl n or certain lie subgroups of the cremona group of birational transformations in n variables quispel et al demonstrated the use of canonical group variables for achieving this reduction we develop canonical coordinates for several groups with elementary lie algebras and demonstrate reduction of order in each case results are used to reduce the order of several examples of recursion formulae taken from the literature on renormalizable lattice models

statistical mechanics a perspective on the work of r j baxter abstract the work of rodney baxter in statistical mechanics is surveyed with an emphasis on the role that his exact calculations have played in the development of the modern theory of critical phenomena

phase diagram of a six state chiral potts model abstract we study the phase diagram of an isotropic six state chiral potts model on a square lattice by means of both exact and numerical methods the phase diagram of this model presents many similarities with the phase diagrams of the ashkin teller model or the models studied by zamolodchikov and monarstirskii a remarkable line globally invariant under a transformation generalizing the kramers wannier duality seems to correspond to a first order transition line up to a bifurcation point where this line splits into two second order lines all the numerical calculations are compared with exact results which can be performed using a canonical elliptic parametrization of this model the bifurcation point is found to correspond to the intersection of a generalized self dual line with an algebraic curve this curve corresponds to the set of points of the phase diagram for which a non trivial infinite symmetry group of the model degenerates into a finite group of order six the agreement between numerical and analytical results is very good

thermal transmissivity in discrete spin systems formulation and applications thermal transmissivity a notion which arises empirically and leads to considerable simplification of analyses of discrete spin systems is of fundamental interest in graphical and real space renormalization group approaches the authors present a formulation of the theory of thermal transmissivity they show that the situation is different depending on whether the interaction matrices commute in the commuting case the thermal transmissivity is given by the eigenvalues of the interaction matrix while in the non commuting case it is given by the eigenvalues as well as matrices which block diagonalize the interaction matrix the meaning of the thermal transmissivity in diagrammatic analyses of spin models is also elucidated and their results are illustrated by examples finally they present a general formulation of disorder solutions for spin models in terms of thermal transmissivities

the discrete chazy iii system of labrunie conte is not integrable we analyse the discrete form of the chazy iii equation proposed by labrunie and conte with the help of two different integrability criteria singularity confinement and algebraic entropy we show that for all values of the free parameter this third order mapping fails both criteria and thus cannot be integrable

on the complexity of some birational transformations using three different approaches we analyse the complexity of various birational maps constructed from simple operations inversions on square matrices of arbitrary size the first approach comprises the study of the images of lines and relies mainly on univariate polynomial algebra the second approach is a singularity analysis and the third method is more numerical using integer arithmetics these three methods have their own domain of application but they give corroborating results and lead us to a conjecture on the algebraic entropy of a class of maps constructed from matrix inversions

thermodynamic limit for a spin lattice an integrable spin lattice is a higher dimensional generalization of integrable spin chains in this paper we consider a special spin lattice related to quantum mechanical interpretation of the three dimensional lattice model in statistical mechanics zamolodchikov and baxter the integrability means the existence of a set of mutually commuting operators expressed in the terms of local spin variables the significant difference between spin chain and spin lattice is that the commuting set for the latter is produced by a transfer matrix with two equitable spectral parameters there is a specific bilinear functional equation for the eigenvalues of this transfer matrix the spin lattice is investigated in this paper in the limit when both sizes of the lattice tend to infinity the limiting form of bilinear equation is derived it allows to analyze the distributions of eigenvalues of the whole commuting set the ground state distribution is obtained explicitly a structure of excited states is discussed

a classification of four state spin edge potts models we classify four state spin models with interactions along the edges according to their behaviour under a specific group of symmetry transformations this analysis uses the measure of complexity of the action of the symmetries in the spirit of the study of discrete dynamical systems on the space of parameters of the models and aims at uncovering solvable ones we find that the action of these symmetries has low complexity polynomial growth zero entropy we obtain natural parametrizations of various models among which an unexpected elliptic parametrization of the four state chiral potts model which we use to localize possible integrability conditions associated with high genus curves

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

norm bounds for summation of two normal matrices abstract a sharp upper bound is obtained for a i b where a and b are n n hermitian matrices satisfying a 1 i a a 2 i and b 1 i b b 2 i similarly an optimal bound is obtained for u v where u and v are n n unitary matrices with any specified spectra the study leads to some surprising phenomena of discontinuity concerning the spectral variation of unitary matrices moreover it is proven that for two non commuting normal matrices a and b with spectra a and b the optimal norm bound for a b equals min c max a max b extensionsof the results to infinite dimensional cases are also considered

remarks on the quantum dilogarithm a quantum analogue of the dilogarithm function has been introduced by faddeev and kashaev 1994 in such a way that a certain identity in the weyl algebra wq plays the role of the five term dilogarithm identity we study this identity in the limit when q approaches a root of unity and show that it then reduces to the restricted star triangle relation which has been used previously by bazhanov and baxter 1992 as a local integrability condition of a class of three dimensional solvable lattice models

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

tetrahedron equations and the relativistic s matrix of straight strings in 2 1 dimensions the quantums matrix theory of straight strings infinite one dimensional objects like straight domain walls in 2 1 dimensions is considered thes matrix is supposed to be purely elastic and factorized the tetrahedron equations which are the factorization conditions are investigated for the special two colour model the relativistic three strings matrix which apparently satisfies this tetrahedron equation is proposed

the road to the discrete analogue of the painleve property nevanlinna meets singularity confinement the question of integrability of discrete systems is analysed in the light of the recent findings of ablowitz et al who have conjectured that a fast growth of the solutions of a differ ence equation is an indication of nonintegrability the study of the behaviour of the solutions of a mapping is based on the theory of nevanlinna in this paper we show how this approach can be implemented in the csse of second order mappings which include the discrete painlevk equations since the nevanlinna approach does offer only a necessary condition which is not restrictive enough we complement it by the singularity confinement requirement first in an autonomous setting and then for deautonomisation we believe that this three tiered approach is the closest one can get to a discrete analogue of the painlevb property 2003 elsevier science ltd all rights reserved

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

hybrid taguchi chaos of multilevel immune and the artificial bee colony algorithm for parameter identification of chaotic systems in this paper a novel evolutionary learning algorithm is proposed by hybridizing the taguchi method chaos disturbance operation multilevel immune algorithm mia and artificial bee colony algorithm abc the algorithm is thus called htcmiabc to estimate the parameter of chaotic systems the htcmiabc comprises two main different phases first we use the mia as the recognition phase to balance local and global searches and accelerate the search speed to enhance the evolutionary phase second the evolutionary phase is built on the abc and chaos disturbance operation to have the capabilities of exploration and exploitation moreover the taguchi method and crossover operation are inserted between the recognition phase and evolutionary phase for the recombination and diversification of several antibodies to improve the searching ability finally the htcmiabc algorithm is examined by parameter identification of the nonlinear chaotic system simulation results show that the proposed algorithm is more efficient than some typical existing algorithms the effects of noise and population size are investigated as well

corner transfer matrices of the chiral potts model ii the triangular lattice we consider a two dimensional edge interaction model satisfying the star triangle relations for the triangular lattice the corner transfer matrices are functions of three rapidities we show that they possess various factorization properties and satisfy certain equations we indicate how these equations can be solved for the ising model we then consider the three state chiral potts model and obtain low temperature solutions to the equations the conjectured formula for the order parameter the spontaneous magnetization is verified to one more order in a series expansion

a solvable lattice model and related rogers ramanujan type identities an exactly solvable lattice model is presented which is a natural generalization of baxter s hard hexagon model to three spin states the free energy and spin densities are calculated related rogers ramanujan type identities are also studied

the emerging role of number theory in exactly solvable models in lattice statistical mechanics abstract we review the baxter model the hard hexagon model and their multistate generalizations from a point of view that stresses the connection with modular functions and additive number theory it is shown for example that various physical quantities in the hard hexagon model are all expressible in terms of modular functions the use of rogers ramanujan type identities in solvable models is also reviewed

a labelling scheme for higher dimensional simplex equations we present a succinct way of obtaining all possible higher dimensional generalizations of the quantum yang baxter equation qybe using the scheme we could generate the two popular three simplex equations namely the zamolodchikov tetrahedron equation and the frenkel and moore equation

cartesian decompositions and schatten norms abstract let t be a hilbert space operator with t a i b where a and b are hermitian we prove sharp inequalities comparing the norms t p with a 2 b 2 1 2 p and a 2 p b 2 p 1 2

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

zamolodchikov s tetrahedron equation and hidden structure of quantum groups the tetrahedron equation is a three dimensional generalization of the yang baxter equation its solutions define integrable three dimensional lattice models of statistical mechanics and quantum field theory their integrability is not related to the size of the lattice therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices obviously any such three dimensional model can be viewed as a two dimensional integrable model on a square lattice where the additional third dimension is treated as an internal degree of freedom therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this hidden third dimension in this paper we construct a new solution of the tetrahedron equation which provides in this way the two dimensional solvable models related to finite dimensional highest weight representations for all quantum affine algebra where the rank n coincides with the size of the hidden dimension these models are related to an anisotropic deformation of the sl n invariant heisenberg magnets they were extensively studied for a long time but the hidden 3d structure was hitherto unknown our results lead to a remarkable exact rank size duality relation for the nested bethe ansatz solution for these models note also that the above solution of the tetrahedron equation arises in the quantization of the resonant three wave scattering model which is a well known integrable classical system in 2 1 dimensions

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

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

investigating the integrability of the lyness mappings we examine a family of difference equations known as the lyness mappings from the point of view of integrability we show that the mappings satisfy two different integrability criteria and are thus good integrability candidates we introduce an ansatz which reduces the mappings to bilinear form and we show that the equations obtained are just reductions of the hirota miwa equation which establishes the integrable character of the lyness mappings finally we discuss the construction of explicit invariants for some instances of the mapping

some constant solutions to zamolodchikov s tetrahedron equations the author presents constant solutions to the tetrahedron equations proposed by zamolodchikov 1981 in general from a given solution of the yang baxter equation there are two ways to construct solutions to the tetrahedron equation there are also other kinds of solutions they present some two dimensional solutions that were obtained by directly solving the equations using either an upper triangular or zamolodchikov s ansatz

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

on solving the ising model functional relation without elliptic functions the exactly solvable n state chiral potts model differs from previous solvable lattice models in that its mathematics leads to hyperelliptic rather than ordinary elliptic functions it is still an open question whether these hyperelliptic functions provide a useful computational tool in particular it has recently been shown that the functional relation for the eigenvalues of the transfer matrix can be solved directly without introducing such functions when n 2 the model reduces to the ising model and this method provides a novel and straightforward way of solving the ising model in fact the method simplifies considerably here we present the resulting technique

the six and eight vertex models revisited elliott lieb s ice type models opened up the whole field of solvable models in statistical mechanics here we discuss the commuting transfer matrix t qequations for these models writing them in a more explicit and transparent notation that we believe offers new insights the approach manifests the relationship between the six vertex and chiral potts models and between the eight vertex and kashiwara miwa models

chaotic bayesian optimal prediction method and its application in hydrological time series the embedding dimension and the number of nearest neighbors are very important parameters in the prediction of chaotic time series to reduce the prediction errors and the uncertainties in the determination of the above parameters a new chaos bayesian optimal prediction method cbopm is proposed by choosing optimal parameters in the local linear prediction method llpm and improving the prediction accuracy with bayesian theory in the new method the embedding dimension and the number of nearest neighbors are combined as a parameter set the optimal parameters are selected by mean relative error mre and correlation coefficient cc indices according to optimization criteria real hydrological time series are taken to examine the new method the prediction results indicate that cbopm can choose the optimal parameters adaptively in the prediction process compared with several llpm models the cbopm has higher prediction accuracy in predicting hydrological time series

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

integrable mappings from matrix transformations and their singularity properties various examples of bi rational transformations having their origin in the theory of exactly solvable vertex models in lattice statistical mechanics as well as bi rational transformations originating from spin edge models are analysed using the singularity confinement method this method provides results concerning the integrable or not character of these bi rational transformations in complete agreement with the results obtained by visualization methods as well as methods based on the analysis of algebraic invariants

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

from integrability to weak chaos abstract we analyze 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 we concentrate on 4 4 matrices and elementary transpositions of two entries this analysis brings out six classes of birational transformations three classes correspond to integrable mappings their iteration yielding elliptic curves the iterations corresponding to the three other classes are included in higher dimensional non trivial algebraic varieties for many initial conditions in the parameter space these orbits lie on transcendental curves and finally explode in these higher dimensional varieties these transformations act on fifteen or q 2 1 variables however one can associate to them remarkably simple non linear recurrences bearing on a single variable the study of these last recurrences gives a complementary understanding of these amazingly regular non integrable mappings

growth and integrability in discrete systems we introduce a new discrete integrability criterion inspired from the recent findings of ablowitz and collaborators this criterion is based on the study of the growth of some characteristic of the solutions of a mapping using nevanlinna theory since the practical implementation of the latter does not always lead to a clear cut answer we complement the growth criterion by the singularity confinement property this combination turns out to be particularly efficient its application allows us to recover the known forms of the discrete painleve equations and to show that no new ones may exist within a given parametrization

an integral arising from the chiral sl n potts model we show that the integral can be expressed in terms of 5f4 hypergeometric functions the integral arises in the solution by baxter and bazhanov of the free energy of the sl n potts model which includes the term j 2 our result immediately gives the logarithmic mahler measure of the laurent polynomial in terms of the same hypergeometric functions