algebraic invariants of the o 2 gauge transformation the authors consider the o 2 gauge transformation for a two state vertex model on a lattice and derive its fundamental algebraic invariants the minimal set of homogeneous polynomials of the vertex weights which are invariant under o 2 transformations explicit expressions of the fundamental invariants are given for symmetric vertex models on lattices with coordination number p 2 3 4 5 6 generalising p 3 results obtained previously from more elaborate considerations

eisenstein series and string thresholds we investigate the relevance of eisenstein series for representing certain g invariant string theory amplitudes which receive corrections from bps states only g may stand for any of the mapping class t duality and u duality groups sl d so d d or e d 1 d 1 respectively using g invariant mass formulae we construct invariant modular functions on the symmetric space k g of non compact type with k the maximal compact subgroup of g that generalize the standard non holomorphic eisenstein series arising in harmonic analysis on the fundamental domain of the poincare upper half plane comparing the asymptotics and eigenvalues of the eisenstein series under second order differential operators with quantities arising in one and g loop string amplitudes we obtain a manifestly t duality invariant representation of the latter conjecture their non perturbative u duality invariant extension and analyze the resulting non perturbative effects this includes the r 4 and r 4 h 4 g 4 couplings in toroidal compactifications of m theory to any dimension d 4 and d 6 respectively

staggered lattice models some staggered models are studied along three different lines exact solvability yang baxter equations the existence of other exact solutions called disorder solutions their group of symmetries the analysis based on the commutation of row to row transfer matrices of arbitrary size n suggests that the exactly solvable models are either of non staggered type or of free fermion staggered type in contrast disorder solutions can easily be exhibited for staggered models the example of the ashkin teller model is analysed finally the symmetry group of these models is seen to be a straightforward generalisation of the one of homogeneous models

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

diagrammatic construction of an irreducible basis of algebraic invariants for the sixteen vertex model according to a fundamental theorem by hilbert the partition function and other physically relevant properties of the general sixteen vertex model must be expressible algebraically in terms of a small number of basic invariants with respect to the symmetry group of the model the construction of such an irreducible set of algebraic invariants has been described in two earlier papers by graff and hijmans1 2 in the present paper it is shown that this basic set can be obtained in a simpler and more elegant way by means of a diagrammatic technique based on scalar and vectorial multiplication of three kinds of construction elements and the use of six reduction rules for diagrams

free fermion checkerboard and z invariant lattice models in statistical mechanics it is shown that the two dimensional free fermion model is equivalent to a checkerboard ising model which is a special case of the general z invariant ising model expressions are given for the partition function and local correlations in terms of those of the regular square lattice ising model corresponding results are given for the self dual potts model and the application of the methods to the three dimensional zamolodchikov model is discussed the paper ends with a discussion of the critical and disorder surfaces of the checkerboard potts model

mirror symmetry on kummer type k3 surfaces we investigate both geometric and conformal field theoretic aspects of mirror symmetry on n 4 4 superconformal field theories with central charge c 6 our approach enables us to determine the action of mirror symmetry on non stable singular fibers in elliptic fibrations of n orbifold limits of k3 the resulting map gives an automorphism of order 4 8 or 12 respectively on the smooth universal covering space of the moduli space we explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories the classical mckay correspondence allows for a natural interpretation of our results

c 2 rational toroidal conformal field theories via the gauss product we find a concise relation between the moduli of a rational narain lattice and the corresponding momentum lattices of left and right chiral algebras via the gauss product as a byproduct we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices

ising model on nonorientable surfaces exact solution for the mobius strip and the klein bottle closed form expressions are obtained for the partition function of the ising model on an mxn simple quartic lattice embedded on a mobius strip and a klein bottle the solutions all lead to the same bulk free energy but for finite m and n the expressions are different depending on whether the strip width m is odd or even finite size corrections at criticality are analyzed and compared with those under cylindrical and toroidal boundary conditions our results are consistent with the conformal field prediction of a central charge c 1 2 provided that the twisted mobius boundary condition is regarded as a free or fixed boundary

a taxonomy and comparison of parallel block multi level preconditioners for the incompressible navier stokes equations in recent years considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible navier stokes equations based on preconditioned krylov methods these include physics based methods such as simple and purely algebraic preconditioners based on the approximation of the schur complement all these techniques can be represented as approximate block factorization abf type preconditioners the goal is to decompose the application of the preconditioner into simplified sub systems in which scalable multi level type solvers can be applied in this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the navier stokes system first presented in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations computational methods in applied mechanical engineering 188 2000 505 526 this taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain schur complement operators we then present a parallel computational study that examines the performance of these methods and compares them to an additive schwarz domain decomposition dd algorithm results are presented for two and three dimensional steady state problems for enclosed domains and inflow outflow systems on both structured and unstructured meshes the numerical experiments are performed using mpsalsa a stabilized finite element code

exact results for a checkerboard ising model with crossing and four spin interactions a seven parameter ising model with crossing and four spin interactions on a checkerboard lattice is mapped by a duality transformation onto a general eight vertex model which has only five independent parameters in a six dimensional subspace of the ising model for which the corresponding eight vertex model satisfies the free fermion condition an exact solution is found

nonintersecting string model and graphical approach equivalence with a potts model using a graphical method we establish the exact equivalence of the partition function of aq state nonintersecting string nis model on an arbitrary planar even valenced lattice with that of a q2 state potts model on a related lattice the nis model considered in this paper is one in which the vertex weights are expressible as sums of those of basic vertex types and the resulting potts model generally has multispin interactions for the square and kagome lattices this leads to the equivalence of a staggered nis model with potts models with anisotropic pair interactions indicating that these nis models have a first order transition forq 2 for the triangular lattice the nis model turns out to be the five vertex model of wu and lin and it relates to a potts model with two and three site interactions the most general model we discuss is an oriented nis model which contains the six vertex model and the nis models of stroganov and schultz as special cases

numerical simulations of strain localization in inelastic solids using mesh free methods in this paper a comprehensive account on using mesh free methods to simulate strain localization in inelastic solids is presented using an explicit displacement based formulation in mesh free computations high resolution shear band formations are obtained in both two dimensional 2 d and three dimensional 3 d simulations without recourse to any mixed formulation discontinuous incompatible element or special mesh design the numerical solutions obtained here are insensitive to the orientation of the particle distributions if the local particle distribution is quasi uniform which to a large extent relieves the mesh alignment sensitivity that finite element methods suffer moreover a simple h adaptivity procedure is implemented in the explicit calculation and by utilizing a mesh free hierarchical partition of unity a spectral wavelet adaptivity procedure is developed to seek high resolution shear band formations moreover the phenomenon of multiple shear band and mode switching are observed in numerical computations with a relatively coarse particle distribution in contrast to the costly fine scale finite element simulations copyright 2000 john wiley sons ltd

simplified bifurcation analysis of the sixteen vertex model based on the standard representation of equivalence classes of models the search for bifurcation points in the parameter space of sixteen vertex models as described in three papers by gaaff and hijmans1 and hijmans and schram2 3 canbe considerably simplified and clarified by representing the equivalence classes of models having the same partition function by their standard models instead of by their normal models the preference for the standard model arises from the fact that it is characterized by parameters belonging to the irreducible representations of the symmetry group of the partition function whereas the parameters defining the normal model belong to a mixed representation the standard model defined in terms of 10 characteristic parameters is unique apart from a finite stargroup of 96 operations arising from the freedom to permute the eigenvalues and eigenvectors of a characteristic 4 x 4 matrix qss bifurcations occur whenever the set of 10 characteristic parameters is invariant with respect to a non trivial little group of operations from the stargroup

study of a two dimensional fully frustrated model the authors consider a fully frustrated ising model on a square lattice depending on four parameters the partition function is shown to be equivalent to the two parameter ferromagnetic onsager partition function this result generalises a relation established by southern et al 1980 and can be checked in the particular case of the one parameter villain model in particular the residual entropy of the villain model is linked to the free energy of the onsager model at criticality the reduction from four to two parameters occurring in this mapping is studied in the light of the inverse relation satisfied by the partition function of the frustrated model

grassman path integral solution for a class of triangular type decorated ising models abstract grassman path integral solution is given for a class of two dimensional triangular type decorated ising models canonical ising lattices rectangular triangular and hexagonal enter into this class as the most simple particular cases as a first step the problem is reformulated in terms of a free fermionic field theory the method is based on the mirror factorization principle for the density matrix traditional transfer matrix or combinatorial considerations are not needed the solution exhibits the characteristics free fermionic structure providing the universal logarithmic singularity in the specific heat the symmetries and the critical point behaviour are investigated within the spin polynomial interpretation of the problem some concrete decorated lattices are treated by illustration attention is given to the choice of rational computational devices

improving the run time and quality of nested dissection ordering when performing sparse matrix factorization the ordering of matrix rows and columns has a dramatic impact on the factorization time this paper describes an approach to the reordering problem that produces significantly better orderings than prior methods the algorithm is a hybrid of nested dissection and minimum degree ordering and combines an assortment of different algorithmic advances new or improved algorithms are described for graph compression multilevel partitioning and separator improvement when these techniques are combined the resulting orderings average 39 better than minimum degree over a suite of test matrices while requiring roughly 2 7 times the run time of liu s multiple minimum degree

on the bifurcations occurring in the parameter space of the sixteen vertex model ii bifurcations arising from degeneracies in the n matrix the analysis of the bifurcation structure in the 16 dimensional parameter space of homogeneous sixteen vertex models started in part i is completed in this paper as before equivalence classes of models having the same partition function are identified by means of a 4 4 diagonal matrix n and a pair of characteristic 2 2 matrices a and b the bifurcation classes of models studied in this part include also classes for which the n matrix shows degeneracies the four primary bifurcation classes uncovered in part i i e the general and complementary eight vertex models and the one sided and doubly cyclic models each give rise to a hierarchy of two kinds of subbifurcations on the one hand the models corresponding to the different patterns of degeneracies of the n matrix and on the other hand the models for which the rotation angles and characterizing the matrices a and b have values 2 na and 2 nb respectively with na and nb integers

a note on preconditioning nonsymmetric matrices the preconditioners for indefinite matrices of kkt form in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 are extended to general nonsymmetric matrices

accurate determination of the critical line of the square ising antiferromagnet in a field several points on the critical line of the simple quadratic antiferromagnetic ising model in a field are located with high precision this is accomplished by means of finite size scaling of the magnetic correlation length of n infinity strips the latter quantities were determined by means of the transfer matrix method for strip widths up to n 20 an expression possibly with the exact form is constructed which reproduces the critical line with an accuracy to the order of 10 10

on the solutions of the yang baxter equations with general inhomogeneous eight vertex r matrix relations with zamolodchikov s tetrahedral algebra we present most general one parametric solutions of the yang baxter equations ybe for one spectral parameter dependent r ij u matrices of the six and eight vertex models where the only constraint is the particle number conservation by mod 2 a complete classification of the solutions is performed we have obtained also two spectral parameter dependent particular solutions r ij u v of ybe the application of the non homogeneous solutions to construction of zamolodchikov s tetrahedral algebra is discussed

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 generalised six vertex model gives conditions under which a certain very general version of the ice model of lattice statistics can be solved special cases of this model include the eight vertex model the three colour problem and the staggered ice model soluble by the pfaffian method all of these special cases are shown to satisfy the authors criteria of solubility

on the bifurcations occuring in the parameter space of the sixteen vertex model i discrete bifurcations abstract the 16 dimensional parameter space of homogeneous sixteen vertex models is scanned for bifurcation points i e points corresponding to models which possess extra symmetries not existing in nearby points equivalence classes of models having the same partition function are identified by means of a characteristic normal model represented by a 4 x 4 diagonal matrix n and a pair of 2 x 2 matrices a and b in this paper the matrix n is assumed to be non degenerate and the only bifurcations found are those associated with special types of matrices a and b i e matrices whose decomposition in terms of pauli matrices corresponds to a vector a a 1 a 2 a 3 or b b 1 b 2 b 3 that is invariant with respect to one or more elements of the cubic symmetry group the various bifurcation classes of models found include a the families of general and complementary eight vertex models b discrete sets of doubly and one sided cyclic models and c a number of secondary bifurcation classes within the eight vertex families among which is baxters symmetric eight vertex model

a semi smooth newton method for orthotropic plasticity and frictional contact at finite strains abstract a new approach for the unified treatment of frictional contact and orthotropic plasticity at finite strains using semi smooth newton methods is presented the contact discretization is based on the well known mortar finite element method using dual lagrange multipliers to facilitate the handling of the additional lagrange multiplier degrees of freedom exploiting the similarity of the typical inequality constraints of plasticity and friction all involved discrete inequalities are reformulated as nonlinear non smooth equations using complementarity functions the resulting set of discrete semi smooth equations can be solved efficiently by a variant of newton s method where all additionally introduced variables are condensed from the global system so that a linear system only consisting of the displacement degrees of freedom has to be solved in each iteration step in contrast to classical radial return mapping methods for computational plasticity the plastic constraints are not required to hold at every iterate in the nonlinear solution procedure but only at convergence this relaxation in the pre asymptotic behavior results in an increased flexibility regarding algorithm design and a potentially higher robustness compared to radial return mapping algorithms the presented elasto plasticity algorithm includes arbitrary isotropic hyperelasticity an anisotropic hill type yield function with isotropic and kinematic hardening plastic spin and appropriate finite element technology for nearly incompressible materials therefore it is well suited for the modeling of sheet metal forming and similar processes several numerical examples underline the robustness of the proposed plasticity algorithm and the efficient treatment of elasto plastic contact problems

fa yueh wu s contributions in physics abstract we sketch a subset of professor f y wu s contributions in lattice statistical mechanics solid state physics graph theory enumerative combinatorics and other domains of physics and mathematics we will recall some of f y wu s most important and well known classic results and we will also sketch some of his most recent researches dedicated to the connections of lattice statistical mechanical models with deep problems in pure mathematics

on complex zeros of the q potts partition function for a self dual family of graphs this paper deals with the location of the complex zeros of q potts partition function for a class of self dual graphs for this class of graphs as the form of the eigenvalues is known the regions of the complex plane can be focused on the sets where there is only one dominant eigenvalue in particular containing the positive half plane thus in these regions the analyticity of the free energy per site can be derived easily next some examples of graphs with their tutte polynomial having few eigenvalues are given the case of the cycle with an edge having a high order of multiplicity is presented in detail in particular we show that the well known conjecture of chen et al is false in the finite case furthermore we obtain a sequence of self dual graphs for which the unit circle does not belong to the accumulation sets of the zeros

d brane charges in gepner models we construct gepner models in terms of coset conformal field theories and compute their twisted equivariant k theories these classify the d brane charges on the associated geometric backgrounds and therefore agree with the topological k theories we show this agreement for various cases in particular the fermat quintic

self duality of the o 2 gauge transformation and the phase structure of vertex models abstract for the symmetric two state vertex model formulated on a lattice with an arbitrary coordination number q we construct a variational series expansion of the free energy with a free gauge parameter playing the role of the variational variable in the lowest order of the variational series expansion we obtain the bethe approximation its analytical treatment provides a new method of searching for the self dual manifolds for lattices of higher coordination number q and gives some information about the internal structure of the self dual manifolds where the first and second order phase transitions take place the results are systematically improved by considering higher order terms in the variational series expansion

parallel multilevel series k way partitioning scheme for irregular graphs in this paper we present a parallel formulation of a multilevel k way graph partitioning algorithm a key feature of this parallel formulation is that it is able to achieve a high degree of concurrency while maintaining the high quality of the partitions produced by the serial multilevel k way partitioning algorithm in particular the time taken by our parallel graph partitioning algorithm is only slightly longer than the time taken for re arrangement of the graph among processors according to the new partition experiments with a variety of finite element graphs show that our parallel formulation produces high quality partitionings in a short amount of time for example a 128 way partitioning of graphs with one million vertices can be computed in a little over two seconds on a 128 processor cray t3d furthermore the quality of the partitions produced is comparable edge cuts within 5 to those produced by the serial multilevel k way algorithm thus our parallel algorithm makes it feasible to perform frequent repartitioning of graphs in dynamic computations without compromising the partitioning quality

block preconditioners based on approximate commutators this paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible navier stokes equations we consider the pressure convection diffusion preconditioners proposed by kay loghin and wathen siam j sci comput 24 2002 pp 237 256 and silvester elman kay and wathen j comput appl math 128 2001 pp 261 279 numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology a drawback however is that it requires the construction of a convection diffusion operator denoted f p projected onto the discrete pressure space this means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues something often held only by the developers of the model as an alternative we consider automatic ways of computing f p based on purely algebraic considerations the new methods are closely related to the bfbt preconditioner of elman siam j sci comput 20 1999 pp 1299 1316 we use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection diffusion operators together with methods for computing sparse approximate inverses to generate the required matrix f p automatically we demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods given a nonsingular matrix a and a matrix t of the same order under certain very mild conditions there is a unique splitting a b c such that t b 1 c moreover all properties of the splitting are derived directly from the iteration matrix t these results do not hold when the matrix a is singular in this case given a matrix t and a splitting a b c such that t b 1 c there are infinitely many other splittings corresponding to the same matrices a and t and different splittings can have different properties for instance when t is nonnegative some of these splittings can be regular splittings while others can be only weak splittings analogous results hold in the symmetric positive semidefinite case given a singular matrix a not for all iteration matrices t there is a splitting corresponding to them necessary and sufficient conditions for the existence of such splittings are examined as an illustration of the theory developed the convergence of certain alternating iterations is analyzed different cases where the matrix is monotone singular and positive semi definite are studied

eight vertex model and ising model in a non zero magnetic field honeycomb lattice the known equivalence of the honeycomb eight vertex model with an ising model in a non zero magnetic field is derived via a direct mapping compared with a previous derivation which uses the generalised weak graph transformation the new method is simpler and more direct and can be extended to other considerations

the o 3 gauge transformation and 3 state vertex models the authors consider the o 3 gauge transformation for three state vertex models on lattices of coordination number three using an explicit mapping between o 3 and sl 2 they establish that there exist exactly six polynomials of the vertex weights which are fundamentally invariant under the o 3 transformation explicit expressions of these fundamental invariants are obtained in the case of symmetric vertex weights

order and disorder lines in systems with competing interactions iii exact results from stochastic crystal growth the methods presented in the first two articles of this series are simplified and generalized by growing stationary stochastic crystals from a given ansatz layer on the disorder trajectory the free energy correlation functions and multicritical points are calculated explicitly for a large class of models with competing interactions including the staggered eight vertex model the general sixteen vertex model theq state potts model on a triangular lattice a generalz q model and restricted spin glass models in two dimensions

graphical approach to the nonintersecting string model star triangle equation inversion relation and exact solution abstract a general q state nonintersecting string model for q 2 is studied and solved using a graphical approach this model of which special cases were first introduced by strogonov and schultz has recently drawn further attention because of its relation to potts models in its simplest version a uniform separable nonintersecting string model on a quadratic lattice it corresponds to a q2 state potts model with potts spins on every other lattice face we first formulate the related star triangle equation which we solve in terms of a line variable parametrization such a parametrization of a solution of the star triangle equation leads to a simple and direct graphical derivation of an inversion relation for the partition function our graphical analysis also shows that the inversion relation holds if certain boundary effects can be neglected as we shall give a special finite lattice from which the inversion relation can be read off immediately this relation is next solved under appropriate analyticity assumptions again using a simple and direct approach and the result is applied to obtain exact solutions for specific string models

numerical computation of algorithmic consistent tangent moduli in large strain computational inelasticity abstract an algorithm for the numerical computation of so called algorithmic or consistent tangent moduli in finite inelasticity is presented these moduli determine the sensitivity of algorithmic expressions for stresses with respect to the change in total deformation they serve as iteration operators by application of newton type solvers for the iterative solution of non linear initial boundary value problems in finite inelasticity the underlying concept of the numerical computation is a perturbation technique based on a forward difference approximation which reduces the computation of the tangent moduli to a multiple stress computation the algorithmic procedure is material independent and surprisingly simple it is outlined for a lagrangian as well as an eulerian geometric setting and applied to model problems of finite elasticity and finite elastoplasticity

exact results of an ising model with three and five spin interactions on a square lattice abstract we investigate an ising model with three spin interactions and five spin interactions on a square lattice we use exact mapping relations among interaction parameters and the temperature we found that the system is equivalent to the eight vertex model in some cases for the system with an imaginary interaction parameter the system is reduced to the exactly solvable ising model with nearest neighbor two spin interactions and some non nearest neighbor two spin interactions the so called vaks larkin ovchinnikov model

complete integrability in statistical mechanics and the yang baxter equations abstract in this paper we give a differential formulation of the yang baxter equations the concept of exactly solvable for lattice models is related to the frobenius concept of complete integrability these ideas are illustrated in the context of the baxter model

new link invariant from the chiral potts model a new link invariant is obtained using the exactly solvable chiral potts model and a generalized gaussian summation identity the new link invariant is characterized by roots of unity and does not appear to belong to the usual quantum group family of invariants the invariant is given in terms of a matrix associated with the link diagram and can be readily written down for any given link

hodge elliptic genera and how they govern k3 theories the complex hodge elliptic genus and its conformal field theoretic counterpart were recently introduced by kachru and tripathy refining the traditional complex elliptic genus we construct a different so called chiral hodge elliptic genus which is expected to agree with the generic conformal field theoretic hodge elliptic genus in contrast to the complex hodge elliptic genus as originally defined for k3 surfaces x the chiral hodge elliptic genus is shown to be independent of all moduli moreover employing kapustin s results on infinite volume limits it is shown that it agrees with the generic conformal field theoretic hodge elliptic genus of k3 theories while the complex hodge elliptic genus does not this new invariant governs part of the field content of k3 theories supporting the idea that all their spaces of states have a common subspace which underlies the generic conformal field theoretic hodge elliptic genus and thereby the complex elliptic genus mathematically this space is modelled by the sheaf cohomology of the chiral de rham complex of x it decomposes into irreducible representations of the n 4 superconformal algebra such that the multiplicity spaces of all massive representations have precisely the dimensions required in order to furnish the representation of the mathieu group m 24 that is predicted by mathieu moonshine this is interpreted as evidence in favour of the ideas of symmetry surfing which have been proposed by taormina and the author along with the claim that the sheaf cohomology of the chiral de rham complex is a natural home for mathieu moonshine these investigations also imply that the generic chiral algebra of k3 theories is precisely the n 4 superconformal algebra at central charge c 6 if the usual predictions on infinite volume limits from string theory hold true

n 2 supergravity type iib superstrings and algebraic geometry the geometry ofn 2 supergravity is related to the variations of hodge structure for formal calabi yau spaces all known results in this branch of algebraic geometry are easily recovered from supersymmetry arguments this identification has a physical meaning for a type iib superstring compactified on a calabi yau 3 fold we give exact non perturbative results for the string effective lagrangian our geometrical framework suggests a re formulation of the gepner conjecture about 2 2 superconformal theories as the solution to theschottky problem for algebraic complex manifolds having trivial canonical bundle

analysis of block preconditioners for models of coupled magma mantle dynamics this article considers the iterative solution of a finite element discretization of the magma dynamics equations in simplified form the magma dynamics equations share some features of the stokes equations we therefore formulate analyze and numerically test an elman silvester and wathen type block preconditioner for magma dynamics we prove analytically and demonstrate numerically the optimality of the preconditioner the presented analysis highlights the dependence of the preconditioner on parameters in the magma dynamics equations that can affect convergence of iterative linear solvers the analysis is verified through a range of two and three dimensional numerical examples on unstructured grids from simple illustrative problems through to large problems on subduction zone like geometries the computer code to reproduce all numerical examples is freely available as supporting material

a six vertex model as a diffusion problem derivation of correlation functions a cellular automaton which describes diffusion of particles with exclusion in one dimension is shown to be equivalent to a six vertex model on a critical line the arrow arrow correlation function of the six vertex model is calculated exactly on this line using a transfer matrix method

yang baxter equation for the asymmetric eight vertex model in this paper we study in the manner of baxter r j baxter exactly solved models in statistical mechanics academic new york 1982 the possible integrable manifolds of the asymmetric eight vertex model as expected they occur when the boltzmann weights are either symmetric or satisfy the free fermion condition but our analysis clarifies the reason why both manifolds need to share a universal invariant we also show that the free fermion condition implies three distinct classes of integrable models

on correlation functions of two state vertex models on the honeycomb lattice abstract a family of two state vertex models on the honeycomb lattice is solved exactly using a generalized weak graph transformation technique two systems are analyzed in detail the spin flip symmetric model which is integrable in the whole temperature range and the symmetric model solvable at one specific temperature this temperature turns out to be significant from the point of view of edge edge correlations namely they vanish at

elliptic genera and quantum field theory it is shown that in elliptic cohomology as recently formulated in the mathematical literature the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the dirac operator ink theory this leads to several insights concerning both elliptic cohomology and string theory some of the relevant calculations have been done previously by schellekens and warner in a different context

a hiker s guide to k3 aspects of n 4 4 superconformal field theory with central charge c 6 we study the moduli space of n 4 4 superconformal field theories with central charge c 6 after a slight emendation of its global description we find the locations of various known models in the component of associated to k3 surfaces among them are the 2 and 4 orbifold theories obtained from the torus component of here so 4 4 triality is found to play a dominant role we obtain the b field values in direction of the exceptional divisors which arise from orbifolding we prove t duality for the 2 orbifolds and use it to derive the form of purely within conformal field theory for the gepner model 2 4 and some of its orbifolds we find the locations in and prove isomorphisms to nonlinear models in particular we prove that the gepner model 2 4 has a geometric interpretation with fermat quartic target space

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

dimers on a simple quartic net with a vacancy a seminal milestone in lattice statistics is the exact solution of the enumeration of dimers on a simple quartic net obtained by fisher kasteleyn and temperley fkt in 1961 an outstanding related and yet unsolved problem is the enumeration of dimers on a net with vacant sites here we consider this vacant site problem with a single vacancy occurring at certain specific sites on the boundary of a simple quartic net first using a bijection between dimer and spanning tree configurations due to temperley kenyon propp and wilson we establish that the dimer generating function is independent of the location of the vacancy and deduce a closed form expression for the generating function we next carry out finite size analyses of this solution as well as that of the fkt solution our analyses lead to a logarithmic correction term in the large size expansion for the vacancy problem with free boundary conditions a concrete example exhibiting this difference is given we also find the central charge c 2 in the language of conformal field theory for the vacancy problem as versus the value c 1 when there is no vacancy

no free ends method for lattice animals and vertex models with arbitrary number of states abstract a new no free ends method for generating power series expansions for vertex models with arbitrary number of states is described it is based on first order differential equations providing a recurrent relation connecting the free ends part of the series expansion in the n 1 th order with the nth and n 1 th order of the complete expansion the number of necessary no free ends graphs can be further reduced for symmetric systems by eliminating the nodes which have two neighbors and their incident bonds are in different states

using complex variables to estimate derivatives of real functions a method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described numerical examples illustrating the power of the approximation are presented

n 4 characters in gepner models orbits and elliptic genera we review the properties of characters of the n 4 superconformal algebra in the context of a nonlinear sigma model on k3 how they are used to span the orbits and how the orbits produce topological invariants like the elliptic genus we derive the same expression for the k3 elliptic genus using three different gepner models 16 24 and 43 theories detailing the orbits and verifying that their coefficients fi are given by elementary modular functions we also reveal the orbits for the 1322 144 and 1242 theories we derive relations for cubes of theta functions and study the function 1 n z 1 n 6n 1 k q 6n 1 2 24 for k 1 2 3 4

order and disorder lines in systems with competing interactions ii the irf model systems with competing interactions can be often exactly solved on a restricted subspace of the parameter space called an order or disorder trajectory a simple method introduced within the transfer matrix formalism allows for the calculation of the free energy and spin spin correlation functions along the order and disorder lines of the ising model with all possible interactions around a face of the square lattice irf model the general eight vertex model is thoroughly examined and shows full analogy with the quantum spin chain results of the previous paper

duality properties of a general vertex model the authors consider the duality properties of a general vertex model on a lattice in any spatial dimension the analysis is based on a generalised weak graph transformation under which the partition function of the vertex model remains invariant it is shown that the generalised weak graph transformation is self dual for lattice coordination number q 2 3 4 5 6 and they conjecture that the self dual property holds for general q they also obtain the self dual manifold for q 3 4 and it is found that in an ising subspace the manifold coincides with the known ising critical locus

new collective modes of interaction nature in inhomogeneous ising networks we propose a vertex formulation of the ising model with inhomogeneous external field on multiconnected networks possessing a superbond structure the related technique based on gauge degrees of freedom enables us to recognize new collective modes of interaction nature which provide an exact solution of the inverse profile problem and an explicit form of a local free energy functional on an extended magnetization mode space application is made to a square strip

the three colouring problem as a special eight vertex model the equations giving the free energy per site and those governing the bethe ansatz wavefunctions for a solid on solid sos model equivalent to the eight vertex model are shown to be identical to those arising in the three colouring problem for the particular choice of parameters eta ik 3 v 0 using the original baxter notation

block preconditioners for coupled physics problems finite element discretizations of multiphysics problems frequently give rise to block structured linear algebra problems that require effective preconditioners we build two classes of preconditioners in the spirit of well known block factorizations m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 i c f ipsen siam j sci comput 23 2001 pp 1050 1051 and apply these to the diffusive portion of the bidomain equations and the benard convection problem an abstract generalized eigenvalue problem allows us to give application specific bounds for the real parts of eigenvalues for these two problems this analysis is accompanied by numerical calculations with several interesting features one of our preconditioners for the bidomain equations converges in five iterations for a range of problem sizes for benard convection we observe mesh independent convergence with reasonable robustness with respect to physical parameters and offer some preliminary para

mapping of the symmetric vertex model onto the ising model for an arbitrary lattice coordination abstract we investigate the mapping of the two state symmetric vertex model onto the ising model in a field for a lattice with general coordination number q the analysis is based on a combination of decoration and generalized weak graph transformations it is shown that the mapping is restricted to the manifold formed by the intersection of q 3 hypersurfaces in the space of symmetric vertex weights we also derive dual relations among the model parameters and the free energies of the related symmetric vertex and ising models the choices of the energies of the vertex configurations for which the mapping can be performed throughout the entire temperature range are briefly discussed

potts model and graph theory elementary exposition is given of some recent developments in studies of graphtheoretic aspects of the potts model topics discussed include graphical expansions of the potts partition function and correlation functions and their relationships with the chromatic dichromatic and flow polynomials occurring in graph theory it is also shown that the potts model realization of these classic graph theoretic problems provides alternate and direct proofs of properties established heretofore only in the context of formal graph theory

an overlapping domain decomposition preconditioning method for monolithic solution of shear bands metals subjected to high strain rate impact often exhibit a sudden and profound drop in the material s load bearing capability a ductile failure phenomenon known as shear banding shear bands characterized as material instabilities are driven by shear heating and described as narrow regions that have sustained intense plastic deformation and high temperature rise this coupled thermo mechanical localization problem can be formulated as a nonlinear system with two balance equations momentum and energy and two constitutive laws for elasticity and plasticity in our formulation mixed finite elements are used to discretize the equations in space and an implicit finite difference scheme is used to advance the system in time where at every time step a newton type method is used to solve the system monolithically to that end a block jacobian matrix is formed analytically using g teaux derivatives the resulting jacobian is sparse nonsymmetric and its sparsity pattern and eigenvalue content vary with the different stages of shear bands formation consequently posing a significant challenge to iterative solvers to address this issue an overlapping domain decomposition preconditioner that takes into account the physics of shear bands and the domain in which it forms is proposed the key idea is to resolve the shear band domain accurately while only solving the remaining domain approximately with selective updates when necessary the preconditioner is formed on the basis of an additive schwartz method that is applied to a schur complement partition of the system the proposed preconditioner is implemented in parallel and tested on three different benchmark problems as compared against off the shelf solvers we study the effect of hh and kk refinement that are obtained from isogeometric discretizations of the system the overlapping strategy and its parallel performance excellent performance is demonstrated in serial and parallel on all benchmark examples

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

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

blume emery griffiths model on the honeycomb lattice exact results are obtained for a spin 1 system on the honeycomb lattice with the blume emery griffiths hamiltonian kt j i jsisj k i j si2j2 isi2 h si subject to the constraintk ln coshj forj 0 the system behaves like a spin 1 2 ising ferromagnet with the free energy analytic everywhere except at the first order phase boundaryh 0 tanhj 2 e sqrt 3 derivatives of the free energy across this boundary are discontinuous and we obtain the exact expression for the spontaneous magnetization forj 0 the system can be transcribed into an antiferromagnetic spin 1 2 ising model in a real magnetic field and from this equivalence portions of the exact phase boundary are determined

numerical differentiation for local and global tangent operators in computational plasticity in this paper numerical differentiation is applied to integrate plastic constitutive laws and to compute the corresponding consistent tangent operators the derivatives of the constitutive equations are approximated by means of difference schemes these derivatives are needed to achieve quadratic convergence in the integration at gauss point level and in the solution of the boundary value problem numerical differentiation is shown to be a simple robust and competitive alternative to analytical derivatives quadratic convergence is maintained provided that adequate schemes and stepsizes are chosen this point is illustrated by means of some numerical examples

the odd eight vertex model we consider a vertex model on the simple quartic lattice defined by line graphs on the lattice for which there is always an odd number of lines incident at a vertex this is the odd 8 vertex model which has eight possible vertex configurations we establish that the odd 8 vertex model is equivalent to a staggered8 vertex model using this equivalence we deduce the solution of the odd8 vertex model when the weights satisfy a free fermion condition it is found that the free fermion model exhibits no phase transitions in the regime of positive vertex weights we also establish the complete equivalence of the free fermion odd 8 vertex model with the free fermion 8 vertex model solved by fan and wu our analysis leads to several ising model representations of thefree fermion model with pure 2 spin interactions