transfer matrix inverison identities for exactly solvable lattice spin models a new functional equation satisfied by the commuting row to row transfer matrices is derived for the eight vertex model functional equations of the same form are satisfied by hard hexagons magnetic hard squares the self dual potts models the andrews baxter forrester models and others these new functional equations are called inversion identities because they generalize the inversion relation for local transfer matrices it is conjectured that all solvable models satisfying yang baxter equations possess such inversion identities and that in general these functional equations can be solved for the transfer matrix eigenvalues

a moving grid framework for geometric deformable models geometric deformable models based on the level set method have become very popular in the last decade to overcome an inherent limitation in accuracy while maintaining computational efficiency adaptive grid techniques using local grid refinement have been developed for use with these models this strategy however requires a very complex data structure yields large numbers of contour points and is inconsistent with the implementation of topology preserving geometric deformable models tgdms in this paper we investigate the use of an alternative adaptive grid technique called the moving grid method with geometric deformable models in addition to the development of a consistent moving grid geometric deformable model framework our main contributions include the introduction of a new grid nondegeneracy constraint the design of a new grid adaptation criterion and the development of novel numerical methods and an efficient implementation scheme the overall method is simpler to implement than using grid refinement requiring no large complex hierarchical data structures it also offers an extra benefit of automatically reducing the number of contour vertices in the final results after presenting the algorithm we demonstrate its performance using both simulated and real images

a framework for variational grid generation conditioning the jacobian matrix with matrix norms functionals for variational grid generation in which the jacobian matrix and matrix norms play a central role are presented all first order geometric qualities of a structured grid length area volume angle between sides aspect ratio grid alignment are determined by the jacobian matrix of the induced mapping between the logical and physical regions grids with desirable quality can be generated by requiring the jacobian matrix or the corresponding metric tensor to have certain properties functionals can be obtained by integrating over the logical or physical domain a power of the norm of a matrix derived from the jacobian matrix a judicious choice of this derived matrix results in a functional whose minima are mappings with desired features many well known grid generators fit in this framework for example the harmonic generator arises when one drives the metric tensor toward the identity the framework provides insight into what grid qualities these functionals describe three new directional control functionals are proposed and tested

optimal mass transport for higher dimensional adaptive grid generation in this work we describe an approach for higher dimensional adaptive grid generation based on solving the l 2 monge kantorovich problem mkp which is a special case of the classical optimal mass transportation problem two methods are developed for computing the coordinate transformation used to define the grid adaptation for the first method the transformation is determined by solving a parabolic monge ampere equation for a steady state solution for the second method the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the l 2 mkp several numerical experiments are presented to demonstrate the performance of the mkp methods and to compare them with some related adaptive grid methods the experimental results demonstrate that the mkp methods show promise as effective and reliable methods for higher dimensional adaptive grid generation

global flows with invariant gibbs measures for euler and navier stokes two dimensional fluids we construct a family of probability spaces p 0 associated with the euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow t time evolution leavingp globally invariant t is obtained as the limit of galerkin approximations associated with euler equations p is also in invariant measure for a stochastic process associated with a navier stokes equation with viscosity stochastically perturbed by a white noise force

a finite difference moving mesh method based on conservation for moving boundary problems we propose a velocity based moving mesh method in which we move the nodes so as to preserve local mass fractions consequently the mesh evolves to be finer where the solution presents rapid changes naturally providing higher accuracy without the need to add nodes we use an integral approach which avoids altering the structure of the original equations when incorporating the velocity and allows the solution to be recovered algebraically we apply our method to a range of one dimensional moving boundary problems the porous medium equation richards equation and the crank gupta problem we compare our results to exact solutions where possible or to results obtained from other methods and find that our approach can be very accurate 1 relative error with as few as ten or twenty nodes

invariant discretization schemes for the shallow water equations invariant discretization schemes are derived for the one and two dimensional shallow water equations with periodic boundary conditions while originally designed for constructing invariant finite difference schemes we extend the use of difference invariants to allow construction of invariant finite volume methods as well it is found that the classical invariant schemes converge to the lagrangian formulation of the shallow water equations these schemes require redistribution of the grid points according to the physical fluid velocity i e the mesh cannot remain fixed in the course of the numerical integration invariant eulerian discretization schemes are proposed for the shallow water equations in computational coordinates instead of using the fluid velocity as the grid velocity an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level the numerical conservation of energy mass and momentum is evaluated for both the invariant

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

uniqueness of solutions of the stochastic navier stokes equation with invariant measure given by the enstrophy a stochastic navier stokes equation with space time gaussian white noise is considered having as infinitesimal invariant measure a gaussian measure whose covariance is given in terms of the enstrophy pathwise uniqueness for a e initial velocity is proven for solutions having as invariant measure

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

on the spectrum of six vertex corner transfer matrices we consider the eigenstates of the six vertex corner transfer matrix using its generator which is an operator of heisenberg xxz type in the ordered regime we construct a complete set of fundamental eigenstates and some bethe ansatz eigenstates as well in the disordered critical regime the spectrum of the finite size system collapses in the thermodynamic limit and we show how this depends on in m the size of the system and

kolmogorov equations in infinite dimensions well posedness and regularity of solutions with applications to stochastic generalized burgers equations we develop a new method to uniquely solve a large class of heat equations so called kolmogorov equations in infinitely many variables the equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper lyapunov type functions this way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally lipschitz drifts apart from general analytic interest the main motivation is to apply this to uniquely solve martingale problems in the sense of stroock varadhan given by stochastic partial differential equations from hydrodynamics such as the stochastic navier stokes equations in this paper this is done in the case of the stochastic generalized burgers equation uniqueness is shown in the sense of markov flows

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

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

the two dimensional euler equation a statistical study we construct surface type measures on the level sets for the renormalized energy which is an invariant quantity for the two dimensional periodic euler flow and prove the existence of weak solutions living on such level sets other classes of invariant measures for the motion are also introduced

corner transfer matrix of generalized free fermion vertex systems the hamiltonian limit of the corner transfer matrix ctm of a generalized free fermion vertex system of finite size leads to a quantum spin hamiltonian of the particular form hn sigma n 1n 1 n sigma nx sigma n 1x lambda sigma ny sigma n 1y h sigma nz sigma sub n 1z diagonalization may be achieved for all pairs of parameters lambda h with the use of some new elliptic polynomials which extend the class of special polynomials known so far in the context of ctm

bernstein bezier weight adjusted discontinuous galerkin methods for wave propagation in heterogeneous media abstract this paper presents an efficient discontinuous galerkin method to simulate wave propagation in heterogeneous media with sub cell variations this method is based on a weight adjusted discontinuous galerkin method wadg which achieves high order accuracy for arbitrary heterogeneous media 1 however the computational cost of wadg grows rapidly with the order of approximation in this work we propose a bernstein bezier weight adjusted discontinuous galerkin method bbwadg to address this cost by approximating sub cell heterogeneities by a fixed degree polynomial the main steps of wadg can be expressed as polynomial multiplication and l 2 projection which we carry out using fast bernstein algorithms the proposed approach reduces the overall computational complexity from o n 2 d to o n d 1 in d dimensions numerical experiments illustrate the accuracy of the proposed approach and computational experiments for a gpu implementation of bbwadg verify that this theoretical complexity is achieved in practice

l1 uniqueness of regularized 2d euler and stochastic navier stokes equations abstract a regularization of the 2d euler equation with periodic boundary conditions is introduced having the same infinitesimal invariants as the euler equation a flow of measure preserving transformations is constructed on l 1 spaces induced by the gaussian measure with covariance given by the inverse of the enstrophy and it is shown that this flow is the only measure preserving flow inducing a strongly continuous semigroup on the corresponding l 1 space we also prove similar uniqueness results for a corresponding class of regularized stochastic 2d navier stokes equations

corner transfer matrices abstract for two dimensional lattice models with interactions only between nearest and diagonally nearest neighbour spins a well known concept is the row to row transfer matrix less well known is the corner transfer matrix ctm this has some very useful properties if it is normalized so that its largest eigenvalue is unity and the eigenvalues are arranged in numerically decreasing order then each eigenvalue tends to a limit as the lattice becomes large for those models which have been solved exactly notably the ising eight vertex and hard hexagon models this limiting eigenvalue distribution is very simple being basically that of a direct product of two by two matrices from it the order parameter can easily be obtained for all models one can write down formally exact matrix relations for the ctm but the matrices are of infinite size if one uses a representation in which the ctm is diagonal and then truncates these relations to finite size then one obtains a quite accurate approximation the larger the size the greater the accuracy i g enting and i have thereby obtained comparatively long series expansions for the ising model in a field and for the hard squares model

self adjointness of some infinite dimensional elliptic operators and application to stochastic quantization we consider an operator k l in a hilbert space h where l is an ornstein uhlenbeck operator u w 1 4 h and is the invariant measure associated with l we show that k is essentially self adjoint in the space l 2 h where is the gibbs measure dx z 1 e 2u x dx an application to stochastic quantization is given

a moving mesh finite difference scheme that preserves scaling symmetry for a class of nonlinear diffusion problems abstract a moving mesh finite difference scheme based on local conservation is presented for a class of scale invariant second order nonlinear diffusion problems with moving boundaries that a preserves the scaling properties and b is exact at the nodes for initial conditions sampled from similarity solutions details are presented for one dimensional problems the extension to multidimensions is described and the exactness property is confirmed for two radially symmetric moving boundary problems the porous medium equation and a simplistic glacier equation in addition the accuracy of the scheme is also tested for non self similar initial conditions by computing relative errors in the approximate solution in the l norm and the approximate boundary position indicating superlinear convergence

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

corner transfer matrices and corner magnetization for the ising model the authors consider the corner magnetization of an ising square lattice with free edges along a diagonal direction using some remarkable properties of the corner transfer matrix they give a unified treatment for various opening angles the calculation of the magnetization is reduced to the solution of a matrix equation in the thermodynamic limit they construct an explicit form of this equation and discuss its numerical and asymptotic solution and the results obtained thereby

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

design and application of a gradient weighted moving finite element code ii in two dimensions in part i the authors reported on the design of a robust and versatile gradient weighted moving finite element gwmfe code in one dimension and on its application to a variety of pdes and pde systems this companion paper does the same for the two dimensional 2d case these moving node methods are especially suited to problems which develop sharp moving fronts especially problems where one needs to resolve the fine scale structure of the fronts the many potential pitfalls in the design of gwmfe codes and the special features of the implicit one dimensional 1d and 2d codes which contribute to their robustness and efficiency are discussed at length in part i this paper concentrates on issues unique to the 2d case brief explanations are given of the variational interpretation of gwmfe the geometrical mechanical interpretation simplified regularization terms and the treatment of systems a catalog of inner products which occur in gwmfe is given with particular attention paid to those involving second order operators after presenting an example of the 2d phenomenon of grid collapse and discussing the need for long time regularization the paper reports on the application of the 2d code to several nontrivial problems nonlinear arsenic diffusion in the manufacture of semiconductors the drift diffusion equations for semiconductor device simulation the buckley leverett black oil equations for reservoir simulation and the motion of surfaces by mean curvature

uniqueness results for the generators of the two dimensional euler and navier stokes flows the euler and navier stokes equations for an incompressible fluid in two dimensions with periodic boundary conditions are considered concerning the euler equation previous works analyzed the associated first order liouville operator l as a symmetric linear operator in a hilbert space l2 with respect to a natural invariant gaussian measure given by the enstrophy with the domain subspace of cylinder smooth bounded functions and have shown that there exist self adjoint extensions of l for the navier stokes equation with a suitable white noise forcing term the associated second order kolmogorov operator k has been considered on the same domain as the sum of the liouville operator l with the ornstein uhlenbeck operator q corresponding to the stokes operator and the forcing term existence of a c0 semigroup of contraction in l2 with generator extending the operator k has been proven in this paper it is proven that both l and k are bounded by naturally associated positive schrodinger like operators which are essentially self adjoint on a dense subspace of cylinder functions other uniqueness results concerning l respectively k are also given

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

an adaptive grid method and its application to steady euler flow calculations an adaptive remeshing procedure based on a cell volume deformation method is presented starting with an initial grid this method offers direct cell volume control through the specification of the transformation jacobian grid points are moved with appropriate grid velocities so that the specified cell volume distribution can be achieved at the end of the grid movement without adding or removing grid points the grid velocities are determined by solving a scalar poisson equation this method is applied to solving the compressible euler equations computational test cases of transonic flow over an airfoil are presented and demonstrate the desired control of grid sizes across shock waves

kolmogorov equation associated to a stochastic navier stokes equation abstract a direct solution of the kolmogorov equation associated to a stochastic navier stokes equation is given with restriction to two space dimensions and periodic boundary conditions the existence of a variational solution is proved using a special property of the nonlinear operator

corner transfer matrices and lorentz invariance on a lattice abstract the continuum limit of the baxter eight vertex lattice model is the lorentz invariant massive thirring sine gordon field theory here it is shown that the baxter model exhibits a continuous symmetry which is an exact lattice generalization generalization of lorentz invariance and that the corner transfer matrix recently developed by baxter is a lattice boost operator the role of elliptic function parameters as lattice rapidity variables is discussed

a moving mesh method based on the geometric conservation law a new adaptive mesh movement strategy is presented which unlike many existing moving mesh methods targets the mesh velocities rather than the mesh coordinates the mesh velocities are determined in a least squares framework by using the geometric conservation law specifying a form for the jacobian determinant of the coordinate transformation defining the mesh and employing a curl condition by relating the jacobian to a monitor function one is able to directly control the mesh concentration the geometric conservation law an identity satisfied by any nonsingular coordinate transformation is an important tool which has been used for many years in the engineering community to develop cell volume preserving finite volume schemes it is used here to transform the algebraic expression specifying the jacobian into an equivalent differential relation which is the key formula for the new mesh movement strategy it is shown that the resulting method bears a close relation with the lagrangian method advantages of the new approach include the ease of controlling the cell volumes and therefore mesh adaption and a theoretical guarantee for existence and nonsingularity of the coordinate transformation it is shown that the method may suffer from the mesh skewness a consequence resulting from its close relation with the lagrangian method numerical results are presented to demonstrate various features of the new method

optimal mass transport based adaptive mesh method for phase field models of two phase fluid flows in this paper we present an efficient adaptive mesh method for solving phase field model of a mixture of two phase incompressible fluid flows the adaptive mesh is generated by a coordinate transformation that is determined as the solution of the classical problem of the optimal mass transportation also known as mongekantorovich problem mkp the numerical solution of the mkp is computed by taking the gradient of the steady state solution of a parabolic mongeampre equation pmae several numerical experiments are conducted to demonstrate the performance of the pmae method for solving phase field models the numerical results illustrate that the adaptive mesh computed with pmae method captures well the moving interface of the phase field function

conformal invariance and critical spectrum of corner transfer matrices the finite size scaling spectra of ising model corner transfer matrices and their generators are studied at criticality the generators are diagonalised using fermion algebra the low lying eigenvalues given by the zeros of meixner polynomials are equally spaced and collapse like 1 log n for large n as predicted by conformal invariance the asymptotics are obtained using a generalised euler maclaurin summation formula the shift in the largest eigenvalue is given analytically as pi c 6 log n with central charge c 1 2 the spectrum generating functions for both fixed and free boundary conditions are expressed simply in terms of the c 1 2 virasoro characters chi delta q with modular parameter q exp pi log n and conformal dimensions delta 0 1 2 1 16

a study on moving mesh finite element solution of the porous medium equation an adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption the method is based on the moving mesh partial differential equation approach and employs its newly developed implementation the implementation has several improvements over the traditional one including its explicit compact form of the mesh velocities ease to program and less likelihood of producing singular meshes three types of metric tensor that correspond to uniform and arclength based and hessian based adaptive meshes are considered the method shows first order convergence for uniform and arclength based adaptive meshes and second order convergence for hessian based adaptive meshes it is also shown that the method can be used for situations with complex free boundaries emerging and splitting of free boundaries and the porous medium equation with variable exponents and absorption two dimensional numerical results are presented

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

corner transfer matrices for the ising model we consider corner transfer matrices for the regular ising model and show that finding a diagonal representation is a soluble problem in interacting fermions we show how the requirement of having an invertible diagonalising transformation determines which representation must be used and explain why baxter s original solution of the problem did not produce such an invertible transformation

mathematical and numerical analysis of a robust and efficient grid deformation method in the finite element context among a variety of grid deformation methods the method proposed by bochev liao and de la pena methods partial differential equations 12 1996 pp 489 506 cai et al comput math appl 48 2004 pp 1077 1086 and liao and anderson appl anal 44 1992 pp 285 298 is one of the most favorable because it prevents mesh tangling and offers precise control over the element volumes its numerical realization requires only solving a poisson problem and a system of fully decoupled initial value problems many other deformation methods in contrast involve the solution of complicated nonlinear partial differential equations pdes in this article we introduce a generalization of liao s method which allows for generating a desired mesh size distribution for quite arbitrary grids without giving rise to mesh tangling we elaborate on its numerical realization and prove the convergence of our method our results are confirmed by numerical experiments

an analytic approach to existence and uniqueness for martingale problems in infinite dimensions we prove existence and uniqueness for a class of martingale problems in a hilbert space we solve the associated kolmogorov equation and prove that the corresponding semigroup is determined by a kernel of measures if a schauder type regularity is satisfied

jacobian weighted elliptic grid generation variational grid generation techniques are used to derive and analyze a weighted elliptic grid generator that controls the jacobian of the underlying transformation in a least squares sense the euler lagrange equations for the area and volume generators are weighted forms of the well known laplace generator weights are restricted to the class of p matrices to help achieve global invertibility of the map connecting the weights to the jacobian of the map results in an intuitive means of controlling grid spacing area orthogonality and grid line directions examples are given on the unit square to demonstrate point attraction local refinement directional alignment and adaption to a shock

corner transfer matrix of a critical free fermion system in order to diagonalize the generator of the corner transfer matrix of a critical free fermion 8 vertex system with magnetic field the authors derive an integral representation for the eigenvector components of the generator at large n they numerically verify the predictions of conformal invariance for critical systems

dirichlet operators via stochastic analysis abstract we give sufficient conditions for essential self adjointness of symmetric differential operators dirichlet operators associated with classical dirichlet forms given by probability measures on hilbert spaces under an additional assumption of log concavity for the measures we prove the existence of a gap at the lower end of spectrum of the dirichlet operator and the ergodicity of the corresponding semi group we consider also applications to the study of stochastic dynamics in some classical lattice systems