algebraic reduction of the ising model we consider the ising model on a cylindrical lattice of l columns with fixed spin boundary conditions on the top and bottom rows the spontaneous magnetization can be written in terms of partition functions on this lattice we show how we can use the clifford algebra of kaufman to write these partition functions in terms of l by l determinants and then further reduce them to m by m determinants where m is approximately l 2 in this form the results can be compared with those of the ising case of the superintegrable chiral potts model they point to a way of calculating the spontaneous magnetization of that more general model algebraically

paper error propagation properties of recursive least squares adaptation algorithms the numerical properties of implementations of the recursive least squares identification algorithm are of great importance for their continuous use in various adaptive schemes here we investigate how an error that is introduced at an arbitrary point in the algorithm propagates it is shown that conventional ls algorithms including bierman s ud factorization algorithm are exponentially stable with respect to such errors i e the effect of the error decays exponentially the base of the decay is equal to the forgetting factor the same is true for fast lattice algorithms the fast least squares algorithm sometimes known as the fast kalman algorithm is however shown to be unstable with respect to such errors

numerical stability properties of a qr based fast least squares algorithm the numerical stability of a recent qr based fast least squares algorithm is established from a backward stability perspective a stability domain approach applicable to any least squares algorithm constructed from the set of reachable states in exact arithmetic is presented the error propagation question is shown to be subordinate to a backward consistency constraint which requires that the set of numerically reachable variables be contained within the stability domain associated to the algorithm this leads to a conceptually lucid approach to the numerical stability question which frees the analysis of stationary assumptions on the filtered sequences and obviates the tedious linearization methods of previous approaches moreover initialization phenomena and considerations related to poorly exciting inputs admit clear interpretations from this perspective the algorithm under study is proved in contrast to many fast algorithms to be minimal

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

chiral potts rapidity curve descended from six vertex model and symmetry group of rapidities in this paper we present a systematical account of the descending procedure from the six vertex model to the n state chiral potts model through fusion relations of j operators following the works of bazhanov stroganov and baxter bazhanov perk a careful analysis of the descending process leads to the appearance of the high genus curve as the rapidity constraint for the chiral potts models full symmetries of the rapidity curve are identified as is its symmetry group structure by normalized transfer matrices of the chiral potts model the 2 t relation can be reduced to functional equations over a hyperelliptic curve associated with rapidities by which the degeneracy of 2 eigenvalues is revealed in the case of the superintegrable chiral potts model

rotation based rls algorithms unified derivations numerical properties and parallel implementations this work presents a unified derivation of four rotation based recursive least squares rls algorithms they solve the adaptive least squares problems of the linear combiner the linear combiner without a desired signal the single channel and the multichannel linear prediction and transversal filtering compared to other approaches the authors derivation is simpler and unified and may be useful to readers for better understanding the algorithms and their relationships moreover it enables improvements of some algorithms in the literature in both the computational and the numerical issues all algorithms derived in this work are based on givens rotations they offer superior numerical properties as shown by computer simulations they are computationally efficient and highly concurrent aspects of parallel implementation and parameter identification are discussed

superintegrable chiral potts model thermodynamic properties an inverse model and a simple associated hamiltonian the partition function of then state superintegrable chiral potts model is obtained exactly and explicitly if not completely rigorously for a finite lattice with particular boundary conditions this yields the bulk and surface free energies and horizontal and vertical correlation lengths and interfacial tensions the critical exponents are 1 2 n hor hor 2 n and vert vert 1 and the finite size corrections are obtained at criticality the eigenvalue spectrum of the column to column transfer matrix is that of a direct product ofn byn matrices inverting this matrix gives a related solvable model which is a generalization of the free fermion model the associated hamiltobian has a very simple form suggesting there may be a more direct algebraic method perhaps a generalized clifford algebra for obtaining its eigenvalues

corner transfer matrices in statistical mechanics corner transfer matrices are a useful tool in the statistical mechanics of simple two dimensional models they can be a very effective way of obtaining series expansions of unsolved models and of calculating the order parameters of solved ones here we review these features and discuss the reason why the method fails to give the order parameter of the chiral potts model

functional relations for the order parameters of the chiral potts model following the method of jimbo miwa and others we obtain functional relations for the order parameters of the chiral potts model we have not yet solved these relations here we discuss their properties and show how one should beware of spurious solutions

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

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

the tightness of the kesten stigum reconstruction bound of symmetric model with multiple mutations it is well known that reconstruction problems as the interdisciplinary subject have been studied in numerous contexts including statistical physics information theory and computational biology to name a few we consider a 2q state symmetric model with two categories of q states in each category and 3 transition probabilities the probability to remain in the same state the probability to change states but remain in the same category and the probability to change categories we construct a nonlinear second order dynamical system based on this model and show that the kesten stigum reconstruction bound is not tight when q ge 4

commutation relations of vertex operators for uq sl m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n

free field representation for massive integrable models a new approach to massive integrable models is considered it allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form factors in thesu 2 thirring and sine gordon models

transfer matrix eigenvectors of the baxter bazhanov stroganov τ2 model for n 2 we find a representation of the row to row transfer matrix of the baxter bazhanov stroganov 2 model for n 2 in terms of an integral over two commuting sets of grassmann variables using this representation we explicitly calculate transfer matrix eigenvectors and normalize them it is also shown how form factors of the model can be expressed in terms of determinants and inverses of certain toeplitz matrices

some remarks on a generalization of the superintegrable chiral potts model the spontaneous magnetization of a two dimensional lattice model can be expressed in terms of the partition function w of a system with fixed boundary spins and an extra weight dependent on the value of a particular central spin for the superintegrable case of the chiral potts model with cylindrical boundary conditions w can be expressed in terms of reduced hamiltonians h and a central spin operator s we conjectured in a previous paper that w can be written as a determinant similar to that of the ising model here we generalize this conjecture to any hamiltonians that satisfy a more general onsager algebra and give a conjecture for the elements of s

the critical an 1 1 chain we study the an 1 1 spin chain in the critical regime q 1 we give free boson realizations of the vertex operators and their duals using these free boson realizations we give integral representations for the correlation functions

identities in the superintegrable chiral potts model we present proofs for a number of identities that are needed to study the superintegrable chiral potts model in the q 0 sector

eigenvectors in the superintegrable model i sl2 generators in order to calculate correlation functions of the chiral potts model one only needs to study the eigenvectors of the superintegrable model here we start this study by looking for eigenvectors of the transfer matrix of the periodic 2 tq model which commutes with the chiral potts transfer matrix we show that the degeneracy of the eigenspace of 2 tq in the q 0 sector is 2r with r n 1 l n when the size of the transfer matrix l is a multiple of n we introduce chiral potts model operators different from the more commonly used generators of quantum group from these we can form the generators of a loop algebra for this algebra we then use the roots of the drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra

exactly solved models on planar graphs with vertices in z 3 it is shown how exactly solved edge interaction models on the square lattice may be extended onto more general planar graphs with edges connecting a subset of next nearest neighbour vertices of mathbb z 3 this is done by using local deformations of the square lattice that arise through the use of the star triangle relation similar to baxter s z invariance property these local deformations leave the partition function invariant up to some simple factors coming from the star triangle relation the deformations used here extend the usual formulation of z invariance by requiring the introduction of oriented rapidity lines which form directed closed paths in the rapidity graph of the model the quasi classical limit is also considered in which case the deformations imply a classical z invariance property as well as a related local closure relation for the action functional of a system of classical discrete laplace equations

the inversion relation method for obtaining the free energy of the chiral potts model abstract we derive the free energy of the chiral potts model by the infinite lattice inversion relation method this method is non rigorous in that it always needs appropriate analyticity assumptions guided by previous calculations based on exact finite lattice functional relations we find that in addition to the usual assumption that the free energy be analytic and bounded in some principal domain of the rapidity parameter space that includes the physical regime we also need a much less obvious symmetry we can then obtain the free energy by wiener hopf factorization in the complex planes of appropriate variables together with the inversion relation this symmetry relates the values of the free energy in all neighbouring domains to those in the principal domain

difference equations for the correlation functions of the eight vertex model the authors propose that the correlation functions of the inhomogeneous eight vertex model in the anti ferroelectric regime satisfy a system of difference equations with respect to the spectral parameters solving the simplest difference equation they obtain the expression for the spontaneous staggered polarization conjectured by baxter and kelland 1974 they also discuss a related construction of vertex operators on the lattice

spin operator matrix elements in the superintegrable chiral potts quantum chain we derive spin operator matrix elements between general eigenstates of the superintegrable n symmetric chiral potts quantum chain of finite length our starting point is the extended onsager algebra recently proposed by baxter for each pair of spaces onsager sectors of the irreducible representations of the onsager algebra we calculate the spin matrix elements between the eigenstates of the hamiltonian of the quantum chain in factorized form up to an overall scalar factor this factor is known for the ground state onsager sectors for the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters for the ising quantum chain in a transverse field n 2 case the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the separation of variables method

bifurcations under weak noise dynamical systems driven by gaussian white noise share some general asymptotic properties in the limit of weak noise which are reminiscent of equilibrium thermodynamics these properties and applications to bifurcating systems are discussed

series expansions from the corner transfer matrix renormalization group method the hard squares model the corner transfer matrix renormalization group method is an efficient method for evaluating physical quantities in statistical mechanical models it originates from baxter s corner transfer matrix equations and method and was developed by nishino and okunishi in 1996 in this paper we review and adapt this method previously used for numerical calculations to derive series expansions we use this to calculate 92 terms of the partition function of the hard squares model we use the resulting series to provide evidence supporting the claim that the method is subexponential in the number of generated terms and briefly analyse the singularities of the function

bifurcations in fluctuating systems the center manifold approach bifurcations in fluctuating dynamical systems are studied using the ideas of center manifold reduction the method provides not only a systematic procedure for the reduction of the system to a small number of variables but also a classification scheme for the different kinds of dynamical behavior possible near bifurcation points the joint probability density factorizes into a stationary gaussian densityp v u in the fast variablesv and a time dependent densityp u f in the slow variablesu describing the dynamics on the center manifoldv v0 u p u t obeys a reduced fokker planck equation that can be written in a normal form by means of local nonlinear transformations both additive and multiplicative white noise are considered as is colored noise the results extend and formalize haken s concept of adiabatic elimination of fast variables

on the duality between fast qr methods and lattice methods in least squares adaptive filtering the authors show that fast qr methods and lattice methods in least squares adaptive filtering are duals and follow from identical geometric principles whereas the lattice methods compute the residuals of a projection operation via the forward and backward prediction errors the qr methods compute instead the weights used in the projections within this framework the parameter identification problem is solved using fast qr methods by showing that the reflection coefficients and tap parameters of a least squares lattice filter operating in the joint process mode are immediately available as internal variables in the fast qr algorithms this parameter set can be readily exploited in system identification signal analysis and linear predictive coding for example the relations derived also lead to a fast least squares algorithm of minimal complexity that is a hybrid between a qr and a lattice algorithm the algorithm combines the order recursive properties of the lattice approach with the robust numerical behavior of the qr approach

a conjecture for the superintegrable chiral potts model we adapt our previous results for the partition function of the superintegrable chiral potts model with open boundaries to obtain the corresponding matrix elements of e h where h is the associated hamiltonian the spontaneous magnetization r can be expressed in terms of particular matrix elements of e hs1re h where s1 is a diagonal matrix we present a conjecture for these matrix elements as an m by m determinant where m is proportional to the width of the lattice the author has previously derived the spontaneous magnetization of the chiral potts model by analytic means but hopes that this work will facilitate a more algebraic derivation similar to that of yang for the ising model

some hyperelliptic function identities that occur in the chiral potts model we present a number of hyperelliptic function identities that occur in the chiral potts model some of these are relevant to the outstanding problem of solving the functional relations for the order parameter

functional relations for the order parameters of the chiral potts model low temperature expansions abstract this is the third in a series of papers in which we set up and discuss the functional relations for the split rapidity line correlation function in the n state chiral potts model the order parameters of the model can be obtained from this function here we consider the case n 3 and write the equations explicitly in terms of the hyperelliptic functions parametrization we also present four term low temperature series expansions which we hope will cast light on the analyticity properties needed to solve the relations the problem remains unsolved but we hope that this will prove to be a step in the right direction

spontaneous magnetization of the superintegrable chiral potts model calculation of the determinant d pq for the ising model the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant yang did this by calculating the eigenvalues in the large lattice limit montroll potts and ward expressed it as a toeplitz determinant and used szeg s theorem this is almost certainly the route originally travelled by onsager for the corresponding problem in the superintegrable chiral potts model neither approach appears to work here we show that the determinant dpq can be expressed as that of a product of two cauchy like matrices one can then use the elementary exact formula for the cauchy determinant one of course regains the known result originally conjectured in 1989

transfer matrix functional relations for the generalized τ 2 t q model the n state chiral potts model in lattice statistical mechanics can be obtained as a descendant of the six vertex model via an intermediate q or 2 t q model here we generalize this to obtain a column inhomogeneous 2 t q model and derive the functional relations satisfied by its row to row transfer matrix we do not need the usual chiral potts relations between the nth powers of the rapidity parameters a p b p c p d p of each column this enables us to readily consider the case of fixed spin boundary conditions on the left and right most columns we thereby re derive the simple direct product structure of the transfer matrix eigenvalues of this model which is closely related to the superintegrable chiral potts model with fixed spin boundary conditions

the order parameter of the chiral potts model an outstanding problem in statistical mechanics is the order parameter of the chiral potts model an elegant conjecture for this was made in 1983 it has since been successfully tested against series expansions but there is as yet no proof of the conjecture here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy then one can indeed verify the conjecture the method is based on the broken rapidity line approach pioneered by jimbo et al j phys a 26 2199 2210 1993

chiral potts model with skewed boundary conditions we obtain the transfer matrix functional relations for the chiral potts model with skewed boundary conditions and find that they are the same as for periodic boundary conditions but with modified selection rules as a start toward calculating the interfacial tension in general we here evaluate it in a low temperature limit performing a bethe ansatz type calculation finally we specialize the relations to the superintegrable case verifying the ansatz proposed by albertiniet al

fixed point error analysis of the normalized ladder algorithm an attempt is made to analyze the fixed point error performance of the normalized ladder algorithm for autoregressive system identification assuming rounding arithmetic a preliminary simulation study of this algorithm has shown that the bias in the estimated reflection coefficients is much more predominant than the variance of the error in the estimate the study therefore is directed to find a model for predicting the bias in the estimated reflection coefficients the analysis shows that the roundoff errors associated with the square root operations in one of the algorithm equations are mainly responsible for the bias in the estimated reflection coefficients these errors arise because of the normalization procedure that makes the quantities under the square root operations very close to one two main results are presented in the paper 1 a simplified theoretical expression for predicting the average bias in the estimated reflection coefficients at any stage is derived 2 a recursive relation for the average error arising from the finite precision arithmetic in the squared residuals is derived this relation illustrates how the errors made in one stage affect the errors in the succeeding stages simulations are performed to check the theoretical models the experimental results agree very closely with the theoretical predictions

free energy of the three state τ2 tq model as a product of elliptic functions we show that the free energy of the three state t2 tq model can be expressed as products of jacobi elliptic functions the arguments being those of an hyperelliptic parametrization of the associated chiral potts model this is the first application of such a parametrization to the n state chiral potts free energy problem for no2

the bazhanov stroganov model from 3d approach we apply a three dimensional approach to describe a new parametrization of the l operators for the two dimensional bazhanov stroganov bs integrable spin model related to the chiral potts model this parametrization is based on the solution of the associated classical discrete integrable system using a three dimensional vertex satisfying a modified tetrahedron equation we construct an operator which generalizes the bs quantum intertwining matrix s this operator describes the isospectral deformations of the integrable bs model

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 characterization of rapidity curve in the chiral potts model we study the geometry of high genus curves of rapidity variables in chiral potts model in terms of symmetries we characterize these rieman surfaces and derive their period matrices by the theory of prime forms the temperature like parameter is expressed by hyperelliptic theta functions

a hybrid least squares qr lattice algorithm using a priori errors this paper presents a new minimal and backward stable qr lsl algorithm obtained through the proper interpretation of the system matrix that describes the adaptation and filtering operations of qr rls algorithms the new algorithm is based on a priori prediction errors normalized by the a posteriori prediction error energy as suggested by the interpretation of the system matrix and uses the fact that the latter quantities can be computed via a lattice structure backward consistency and backward stability become guaranteed under simple numerical conventions in contrast with the known a posteriori qr lsl algorithm the new algorithm present fewer numerical complexity and backward consistency is guaranteed without the constraint of passive rotations in the recursive lattice section furthermore reordering of some operations results in a version with identical numerical behavior and inherent parallelism that can be exploited for fast implementations both a priori and a posteriori qr lsl algorithms are compared by means of simulations for small mantissa wordlengths and forgetting factors spl lambda not too close to 1 the proposed algorithm performs better due to dispensing with passive rotations for forgetting factors very close to one and small wordlengths both algorithms are sensitive to the accuracy of some well identified computations

on the stationary distribution of self sustained oscillators around bifurcation points a double expansion in powers of the damping coefficient and noise intensity is shown to be a powerful method for obtaining the stationary distribution of systems that after rescaling become weakly damped conservative ones systems undergoing hopf bifurcations belong to this class as an illustrative example the generalized van der pol oscillator is considered around its bifurcation point a calculation is carried out up to third order in both the noise intensity and the bifurcation parameter damping coefficient

givens rotation based least squares lattice and related algorithms the author presents a general and systematic approach for deriving new ls least squares estimation algorithms that are based solely on givens rotations in particular this approach is used to derive efficient givens rotation based ls lattice algorithms the givens lattice algorithms by exploiting the relationship between the givens algorithms and the recursive modified gram schmidt algorithm it is shown that the time and order update of any order recursive ls estimation algorithm can be realized by employing only givens rotations applying this general conclusion to ls estimation of time series signals results in the givens lattice algorithms two givens lattice algorithms one with square roots and the other without are presented it is shown that the givens lattice algorithms are computationally more efficient than the fast qr algorithm of cioffi 1987 the derivation of other givens rotation based ls estimation algorithms and their systolic array implementations are discussed

extended z invariance for integrable vector and face models and multi component integrable quad equations in a previous paper kels in j phys a 50 49 495202 2017 the author has established an extension of the z invariance property for integrable edge interaction models of statistical mechanics that satisfy the star triangle relation str form of the yang baxter equation ybe in the present paper an analogous extended z invariance property is shown to also hold for integrable vector models and interaction round a face irf models of statistical mechanics respectively as for the previous case of the str the z invariance property is shown through the use of local cubic type deformations of a 2 dimensional surface associated to the models which allow an extension of the models onto a subset of next nearest neighbour vertices of mathbb z 3 while leaving the partition functions invariant these deformations are permitted as a consequence of the respective ybe s satisfied by the models the quasi classical limit is also considered and it is shown that an analogous z invariance property holds for the variational formulation of classical discrete laplace equations which arise in this limit from this limit new integrable 3d consistent multi component quad equations are proposed which are constructed from a degeneration of the equations of motion for irf boltzmann weights

bethe subalgebras in affine birman murakami wenzl algebras and flat connections for q kz equations commutative sets of jucys murphy elements for affine braid groups of types were defined construction of r matrix representations of the affine braid group of type and its distinguished commutative subgroup generated by the type jucys murphy elements are given we describe a general method to produce flat connections for the two boundary quantum knizhnik zamolodchikov equations as necessary conditions for sklyanin s type transfer matrix associated with the two boundary multicomponent zamolodchikov algebra to be invariant under the action of the type jucys murphy elements we specify our general construction to the case of the birman murakami wenzl algebras bmw algebras for short as an application we suggest a baxterization of the dunkl cherednik elements in the double affine hecke algebra of type a

the riemann surface of the chiral potts model free energy function in a recent paper we derived the free energy or partition function of the n state chiral potts model by using the infinite lattice inversion relation method together with a non obvious extra symmetry this gave us three recursion relations for the partition function per site t pq of the infinite lattice here we use these recursion relations to obtain the full riemann surface of t pq in terms of the t p t q variables it consists of an infinite number of riemann sheets each sheet corresponding to a point on a 2n 1 dimensional lattice for n 2 the function t pq is meromorphic on this surface we obtain the orders of all the zeros and poles for n odd we show that these orders are determined by the usual inversion and rotation relations without the extra symmetry together with a simple linearity ansatz for n even this method does not give the orders uniquely but leaves only n 4 4 parameters to be determined

baxter s solution for the free energy of the chiral potts model in honor of baxter s sixtieth birthday we would like to review some of his work on the free energy of the chiral potts model in spite of the enormous complexity and difficulty of the problem baxter using functional relations was able to calculate not only the free energy but also the interfacial tension we here show that the integral for the free energy simplifies in the superintegrable case and is identical to his earlier results using entirely different approaches his calculations are extended to include other regions we also attempt to clarify some of his reasoning as several steps may be mysterious at first glance

vertex operators in solvable lattice models the basic properties of q vertex operators are formulated in the context of the andrews baxter forrester abf series as an example of face interaction models the q difference equations satisfied by their correlation functions are derived and their connection with representation theory established the q difference equations of the kashiwara miwa km series are discussed as an example of edge interaction models next the ising model the simplest special case of both abf and km series is studied in more detail using the jordan wigner fermions in particular all matrix elements of vertex operators are calculated

corner transfer matrices of the chiral potts model we present some symmetry and factorization relations satisfied by the corner transfer matrices ctms of the chiral potts model we show how the single spin expectation values can be expressed in terms of the ctms and in terms of the related boost operator low temperature calculations lead naturally to the variables that uniformize the boltzmann weights of the model

the baxter revolution i review the revolutionary impact rodney baxter has had on statistical mechanics beginning with his solution of the 8 vertex model in 1971 and the invention of corner transfer matrices in 1976 to the creation of the rsos models in 1984 and his continuing current work on the chiral potts model