auxiliary matrices for the six vertex model and the algebraic bethe ansatz we connect two alternative concepts of solving integrable models baxter s method of auxiliary matrices or q operators and the algebraic bethe ansatz the main steps of the calculation are performed in a general setting and a formula for the bethe eigenvalues of the q operator is derived a proof is given for states which contain up to three bethe roots further evidence is provided by relating the findings to the six vertex fusion hierarchy for the xxz spin chain we analyse the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity

new q matrices and their functional equations for the eight vertex model at elliptic roots of unity the q matrix invented by baxter in 1972 to solve the eight vertex model at roots of unity exists for all values of n the number of sites in the chain but only for a subset of roots of unity we show in this paper that a new q matrix which has recently been introduced and is non zero only for n even exists for all roots of unity in addition we consider the relations between all of the known q matrices of the eight vertex model and conjecture functional equations for them

sov approach for integrable quantum models associated with general representations on spin 1 2 chains of the 8 vertex reflection algebra the analysis of the transfer matrices associated with the most general representations of the 8 vertex reflection algebra on spin 1 2 chains is here implemented by introducing a quantum separation of variables sov method which generalizes to these integrable quantum models the method first introduced by sklyanin for representations reproducing in their homogeneous limits the open xyz spin 1 2 quantum chains with the most general integrable boundary conditions we explicitly construct representations of the 8 vertex reflection algebras for which the transfer matrix spectral problem is separated then in these sov representations we get the complete characterization of the transfer matrix spectrum eigenvalues and eigenstates and its non degeneracy moreover we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states which particularly apply to transfer matrix eigenstates a comparison of our analysis of the 8 vertex reflection algebra with that of niccoli g 2012 j stat mech p10025 faldella s et al 2014 j stat mech p01011 for the 6 vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products which exists for these two different realizations of the reflection algebra once they are described by the sov method as will be shown in a future publication this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8 vertex and 6 vertex type

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

fusion operators in the generalized τ 2 model and root of unity symmetry of the xxz spin chain of higher spin we construct the fusion operators in the generalized 2 model using the fused l operators and verify the fusion relations with the truncation identity the algebraic bethe ansatz discussion is conducted on two special classes of 2 which include the superintegrable chiral potts model we then perform the parallel discussion on the xxz spin chain at roots of unity and demonstrate that the sl2 loop algebra symmetry exists for the root of unity xxz spin chain with a higher spin where the evaluation parameters for the symmetry algebra are identified by the explicit fabricius mccoy current for the bethe states parallels are also drawn to the comparison with the superintegrable chiral potts model

the complexity of some polynomial network consistency algorithms for constraint satisfaction problems abstract constraint satisfaction problems play a central role in artificial intelligence a class of network consistency algorithms for eliminating local inconsistencies in such problems has previously been described we analyze the time complexity of several node arc and path consistency algorithms and prove that arc consistency is achievable in time linear in the number of binary constraints the waltz filtering algorithm is a special case of the arc consistency algorithm in the edge labelling computational vision application the constraint graph is planar and so the time complexity is linear in the number of variables

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

the tq equation of the eight vertex model for complex elliptic roots of unity we extend our studies of the tq equation introduced by baxter in his 1972 solution of the eight vertex model with parameter given by 2l 2m1k im2k from m2 0 to the more general case of complex we find that there are several different cases depending on the parity of m1 and m2

a q operator identity for the correlation functions of the infinite xxz spin chain an explicit construction for q operators of the finite xxz spin chain with twisted boundary conditions is presented the massless and the massive regimes are considered as well as the root of unity case it is explained how these results yield an alternative expression for the trace function employed in the description of the correlation functions of the inhomogeneous xxz model on the infinite lattice by boos jimbo miwa smirnov and takeyama

entropy and distance of random graphs with application to structural pattern recognition the notion of a random graph is formally defined it deals with both the probabilistic and the structural aspects of relational data by interpreting an ensemble of attributed graphs as the outcomes of a random graph we can use its lower order distribution to characterize the ensemble to reflect the variability of a random graph shannon s entropy measure is used to synthesize an ensemble of attributed graphs into the distribution of a random graph or a set of distributions we propose a distance measure between random graphs based on the minimum change of entropy before and after their merging when the ensemble contains more than one class of pattern graphs the synthesis process yields distributions corresponding to various classes this process corresponds to unsupervised learning in pattern classification using the maximum likelihood rule and the probability computed for the pattern graph based on its matching with the random graph distributions of different classes we can classify the pattern graph to a class

auxiliary matrices on both sides of the equator the spectra of previously constructed auxiliary matrices for the six vertex model at roots of unity are investigated for spin chains of even and odd length the two cases show remarkable differences in particular it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to baxter s tq equation corresponding to the bethe ansatz equations above and below the equator in contrast one finds for even spin chains only one linear independent solution and complete strings the other main result is the proof of a previous conjecture on the degeneracies of the six vertex model at roots of unity the proof rests on the derivation of a functional equation for the auxiliary matrices which is closely related to a functional equation for the eight vertex model conjectured by fabricius and mccoy

structural matching by discrete relaxation this paper describes a bayesian framework for performing relational graph matching by discrete relaxation our basic aim is to draw on this framework to provide a comparative evaluation of a number of contrasting approaches to relational matching broadly speaking there are two main aspects to this study firstly we focus on the issue of how relational inexactness may be quantified we illustrate that several popular relational distance measures can be recovered as specific limiting cases of the bayesian consistency measure the second aspect of our comparison concerns the way in which structural inexactness is controlled we investigate three different realizations of the matching process which draw on contrasting control models the main conclusion of our study is that the active process of graph editing outperforms the alternatives in terms of its ability to effectively control a large population of contaminating clutter

theoretical analysis of evolutionary algorithms with an infinite population size in continuous space part ii analysis of the diversification role of crossover for pt i see ibid p 102 19 in this part of the paper the authors concentrate on the unique diversification role of the crossover operator in genetic algorithms the explorative behavior of a generic crossover operator is revealed through a detailed large sample analysis recursive equations for the population distributions are derived for a uniform crossover operator in multi dimensional continuous space showing how the crossover operator probes new regions of the solution space while keeping the population within the feasible region the results of this analysis can be extended to the setting of a discrete space in a straightforward manner shedding much light on the understanding of the essential role of crossover in genetic algorithms

auxiliary matrices for the six vertex model at qn 1 ii bethe roots complete strings and the drinfeld polynomial the spectra of recently constructed auxiliary matrices for the six vertex model respectively the spin s 1 2 heisenberg chain at roots of unity are investigated two conjectures are formulated both of which are proved for n 3 and are verified numerically for several examples with n 3 the first conjecture identifies an abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable the zeros of these polynomials are shown to fall into two sets one consists of the solutions to the bethe ansatz equations which determine the eigenvalues of the six vertex transfer matrix the other set of zeros contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry present at roots of 1 the second conjecture then states a polynomial identity which relates the complete string centres to the bethe roots allowing one to determine the dimension of the degenerate eigenspaces its proof for n 3 involves the derivation of a new functional equation for the auxiliary matrices and the six vertex transfer matrix moreover it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the drinfeld polynomial the latter is used to classify the finite dimensional irreducible representations of the loop algebra this suggests that the constructed auxiliary matrices not only enable one to solve the six vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry

conservation errors and convergence characteristics of iterative space marching algorithms the new iterative space marching algorithms for numerical solution of the parabolized navier stokes equations are critically examined it is shown by analysis and numerical examples that employing parabolized flux difference splitting in the marching direction can lead to serious conservation errors even though a finite volume formulation is employed schemes employing parabolized flux vector splitting in the marching direction are shown to be conservative the structure of the iterative algorithm is examined to determine what factors play important roles in the convergence of the local iteration

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

baxter s q operator for the w algebra wn the q oscillator representation for the borel subalgebra of the affine symmetry is presented by means of this q oscillator representation we give the free field realizations of baxter s q operator for the w algebra wn we give functional relations of the t q operators including the higher rank generalization of baxter s t q relation

regular xxz bethe states at roots of unity as highest weight vectors of the sl2 loop algebra we show that every regular bethe ansatz eigenvector of the xxz spin chain at roots of unity is a highest weight vector of the sl2 loop algebra for some restricted sectors with respect to eigenvalues of the total spin operator sz and evaluate explicitly the highest weight in terms of the bethe roots we also discuss whether a given regular bethe state in the sectors generates an irreducible representation or not in fact we present such a regular bethe state in the inhomogeneous case that generates a reducible weyl module here we call a solution of the bethe ansatz equations which is given by a set of distinct and finite rapidities regular bethe roots we call a nonzero bethe ansatz eigenvector with regular bethe roots a regular bethe state

new developments in the eight vertex model ii chains of odd length we study the transfer matrix of the 8 vertex model with an odd number of lattice sites n for systems at the root of unity points mk l with m odd the transfer matrix is known to satisfy the famous tq equation where q is a specifically known matrix we demonstrate that the location of the zeroes of this q matrix is qualitatively different from the case of evenn and in particular they satisfy a previously unknown equation which is more general than what is often called bethe s equation for the case of even m where no q matrix is known we demonstrate that there are many states which are not obtained from the formalism of the sos model but which do satisfy the tq equation the ground state for the particular case of 2k 3 and n odd is investigated in detail

bethe s equation is incomplete for the xxz model at roots of unity we demonstrate for the six vertex and xxz model parameterized by q q 1 2 1 that when q2n 1 for integer n 2 the bethe s ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl2 loop algebra symmetry group of the model therefore in this case the bethe s ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function we discuss how the evaluation parameters of the finite dimensional representations of the sl2 loop algebra can be used to complete this specification

the q operator and functional relations of the eight vertex model at root of unity for odd n following baxter s method of producing q 72 operator we construct the q operator of the root of unity eight vertex model for the crossing parameter 2mk n with odd n where q 72 does not exist we use this new q operator to study the functional relations in the fabricius mccoy comparison between the root of unity eight vertex model and the superintegrable n state chiral potts model by the compatibility of the constructed q operator with the structure of baxter s eight vertex solid on solid sos model we verify the set of functional relations of the root of unity eight vertex model using the explicit form of the q operator and fusion weights of the sos model

the eight vertex model and painleve vi equation ii eigenvector results we study a special anisotropic model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors the results are written in terms of tau functions associated with picard s elliptic solutions of the painleve vi equation connections with other problems related to the eight vertex model are briefly discussed

completing bethe s equations at roots of unity in a previous paper we demonstrated that bethe s equations are not sufficient to specify the eigenvectors of the xxz model at roots of unity for states where the hamiltonian has degenerate eigenvalues we here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the sl 2 loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by bethe s ansatz

a robust evolutionary algorithm for the recovery of rational gielis curves gielis curves gc can represent a wide range of shapes and patterns ranging from star shapes to symmetric and asymmetric polygons and even self intersecting curves such patterns appear in natural objects or phenomena such as flowers crystals pollen structures animals or even wave propagation gielis curves and surfaces are an extension of lame curves and surfaces superquadrics which have benefited in the last two decades of extensive researches to retrieve their parameters from various data types such as range images 2d and 3d point clouds etc unfortunately the most efficient techniques for superquadrics recovery based on deterministic methods cannot directly be adapted to gielis curves indeed the different nature of their parameters forbids the use of a unified gradient descent approach which requires initial pre processings such as the symmetry detection and a reliable pose and scale estimation furthermore even the most recent algorithms in the literature remain extremely sensitive to initialization and often fall into local minima in the presence of large missing data we present a simple evolutionary algorithm which overcomes most of these issues and unifies all of the required operations into a single though efficient approach the key ideas in this paper are the replacement of the potential fields used for the cost function closed form by the shortest euclidean distance sed iterative approach the construction of cost functions which minimize the shortest distance as well as the curve length using r functions and slight modifications of the evolutionary operators we show that the proposed cost function based on sed and r function offers the best compromise in terms of accuracy robustness to noise and missing data

form factors in the baxter bazhanov stroganov model i norms and matrix elements we continue our investigation of the baxter bazhanov stroganov model using the method of separation of variables 1 in this paper we calculate the norms and matrix elements of a local spin operator between eigenvectors of the auxiliary problem for the norm the multiple sums over the intermediate states are performed explicitly in the case n 2 we solve the baxter equation and obtain form factors of the spin operator of the periodic ising model on a finite lattice

properties of the string operator in the eight vertex model the construction of creation operators of exact strings in eigenvectors of the eight vertex model at elliptic roots of unity of the crossing parameter which allow the generation of the complete set of degenerate eigenstates is based on the conjecture that the naive string operator vanishes in this paper we present a proof of this conjecture furthermore we show that for chains of odd length the string operator is either proportional to the symmetry operator s or vanishes depending on the precise form of the crossing parameter

construction of some missing eigenvectors of the xyz spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix we discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight vertex model at the discrete coupling parameters we consider the algebraic bethe ansatz of the elliptic quantum group e sl2 for the case where the parameter satisfies 2n m1 m2 for arbitrary integers n m1 and m2 when m1 or m2 is odd the eigenvectors thus obtained have not been discussed previously furthermore we construct a family of degenerate eigenvectors of the xyz spin chain some of which are shown to be related to the sl2 loop algebra symmetry of the xxz spin chain we show that the dimension of some degenerate eigenspace of the xyz spin chain on l sites is given by n2l n if l n is an even integer the construction of eigenvectors of the transfer matrices of some related interaction round a face models is also discussed

generalized chiral potts models and minimal cyclic representations of u_q widehat germ g germ l n bf c we present for odd n a construction of the n 1 state generalization of the chiral potts model proposed recently by bazhanov et al the yang baxter equation is proved

a metric for comparing relational descriptions relational models are frequently used in high level computer vision finding a correspondence between a relational model and an image description is an important operation in the analysis of scenes in this paper the process of finding the correspondence is formalized by defining a general relational distance measure that computes a numeric distance between any two relational descriptions a model and an image description two models or two image descriptions the distance measure is proved to be a metric and is illustrated with examples of distance between object models a variant measure used in our past studies is shown not to be a metric

a new q matrix in the eight vertex model we construct a q matrix for the eight vertex model at roots of unity for crossing parameter 2mk l with odd l a case for which the existing constructions do not work the new q matrix depends on the spectral parameter v and also on a free parameter t for has the standard properties for t 0 however it does not commute with the operator s nor with itself for different values of the spectral parameter we show that the six vertex limit of exists

inexact graph matching using genetic search this paper describes a framework for performing relational graph matching using genetic search there are three novel ingredients to the work firstly we cast the optimisation process into a bayesian framework by exploiting the recently reported global consistency measure of wilson and hancock as a fitness measure the second novel idea is to realise the crossover process at the level of subgraphs rather than employing string based or random crossover finally we accelerate convergence by employing a deterministic hill climbing process prior to selection since we adopt the bayesian consistency measure as a fitness function the basic measure of relational distance underpinning the technique is hamming distance our standpoint is that genetic search provides a more attractive means of performing stochastic discrete optimisation on the global consistency measure than alternatives such as simulated annealing moreover the action of the optimisation process is easily understood in terms of its action in the hamming distance domain we demonstrate empirically not only that the method possesses polynomial convergence time but also that the convergence rate is more rapid than simulated annealing we provide some experimental evaluation of the method in the matching of aerial stereograms and evaluate its sensitivity on synthetically generated graphs

an antiperiodic dynamical six vertex model i complete spectrum by sov matrix elements of the identity on separate states and connections to the periodic eight vertex model the spin 1 2 highest weight representations of the dynamical six vertex and the standard eight vertex yang baxter algebra on a finite chain are considered in this paper in particular the integrable quantum models associated with the corresponding transfer matrices under antiperiodic boundary conditions for the dynamical six vertex case and periodic boundary conditions for the eightvertexcaseareanalyzedhere fortheantiperiodicdynamicalsix vertextransfer matrix defined on chains with an odd number of sites we adapt sklyanin s quantum separation of variable sov method and explicitly construct the sov representations from the original space of the representations in this way we provide the complete characterization of the eigenvalues and the eigenstates proving also the simplicity of its spectrum moreover we characterize the matrix elements of the identity on separated states of this model by determinant formulae the matrices entering these determinants have elements given by sums over the sov spectrum of the product of the coefficients of the separate states this sov analysis is done without any need to be reduced to the case of the so called elliptic roots of unit and the results derived here define the required setup to extend to the dynamical six vertex model the approach recently developed by the author and collaborators to compute the form factors of the local operators in the sov framework for the periodic eight vertex transfer matrix we prove that its eigenvalues have to satisfy a fixed system of equations in the case of a chain with an odd number of sites this system of equations is the same entering in the sov characterization of the antiperiodic dynamical six vertex transfer matrix spectrum this implies that the set of the periodic eight vertex eigenvalues is contained in the set of the antiperiodic dynamical six vertex eigenvalues a criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate with any of such eigenvalues one nonzero eigenstate of the periodic eight vertex transfer matrix by using the sov results moreover a preliminary discussion

new developments in the eight vertex model we demonstrate that the q matrix introduced in baxter s 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for q v which both contains the bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues

completeness of the bethe ansatz for the six and eight vertex models we discuss some of the difficulties that have been mentioned in the literature in connection with the bethe ansatz for the six vertex model and xxz chain and for the eight vertex model in particular we discuss the beyond the equator infinite momenta and exact complete string problems we show how they can be overcome and conclude that the coordinate bethe ansatz does indeed give a complete set of states as expected

fuzzy morphisms between graphs a generic definition of fuzzy morphism between graphs gfm is introduced that includes classical graph related problem definitions as sub cases such as graph and subgraph isomorphism the gfm uses a pair of fuzzy relations one on the vertices and one on the edges each relation is a mapping between the elements of two graphs these two fuzzy relations are linked with constraints derived from the graph structure and the notion of association graph the theory extends the properties of fuzzy relation to the problem of generic graph correspondence we introduce two complementary interpretations of gfm from which we derive several interesting properties the first interpretation is the generalization of the notion of association compatibility the second is the new notion of edge morphism one immediate application is the introduction of several composition laws each property has a theoretical and a practical interpretation in the problem of graph correspondence that is explained throughout the paper special attention is paid to the formulation of a non algorithmical theory in order to propose a first step towards a unified theoretic framework for graph morphisms

critical evaluation of two equation models for near wall turbulence abstract a variety of two equation turbulence models including several versions of the k epsilon model as well as the k omega model are analyzed critically for near wall turbulent flows from a theoretical and computational standpoint it is shown that the k epsilon model has two major problems associated with it the lack of natural boundary conditions for the dissipation rate and the appearance of higher order correlations in the balance of terms for the dissipation rate at the wall in so far as the former problem is concerned either physically inconsistent boundary conditions have been used or the boundary conditions for the dissipation rate have been tied to higher order derivatives of the turbulent kinetic energy which leads to numerical stiffness the k omega model can alleviate these problems since the asymptotic behavior of w is known in more detail and since its near wall balance involves only exact viscous terms however the modeled form of the omega equation that is used in the literature is incomplete an exact viscous term is missing which causes the model to behave in an asymptotically inconsistent manner by including this viscous term and by introducing new wall damping functions with improved asymptotic behavior a new k tau model where tau 1 omega is turbulent time scale is developed it is demonstrated that this new model is computationally robust and yields improved predictions for turbulent boundary layers jhd

convergence of a hill climbing genetic algorithm for graph matching abstract this paper presents a convergence analysis for the problem of consistent labelling using genetic search the work builds on a recent empirical study of graph matching where we showed that a bayesian consistency measure could be efficiently optimised using a hybrid genetic search procedure which incorporated a hill climbing step in the present study we return to the algorithm and provide some theoretical justification for its observed convergence behaviour the novelty of the analysis is to demonstrate analytically that the hill climbing step significantly accelerates convergence and that the convergence rate is polynomial in the size of the node set of the graphs being matched

comments on theoretical analysis of evolutionary algorithms with an infinite population size in continuous space i basic properties of selection and mutation with reply this paper aims at establishing fundamental theoretical properties for a class of genetic algorithms in continuous space gacs the algorithms employ operators such as selection crossover and mutation in the framework of a multidimensional euclidean space the paper is divided into two parts the first part concentrates on the basic properties associated with the selection and mutation operators recursive formulae for the gacs in the general infinite population case are derived and their validity is rigorously proven a convergence analysis is presented for the classical case of a quadratic cost function it is shown how the increment of the population mean is driven by its own diversity and follows a modified newton s search sufficient conditions for monotonic increase of the population mean fitness are derived for a more general class of fitness functions satisfying a lipschitz condition the diversification role of the crossover operator is analyzed in part ii the treatment adds much light to the understanding of the underlying mechanism of evolution like algorithms