exact solution and interfacial tension of the six vertex model with anti periodic boundary conditions we consider the six vertex model with anti periodic boundary conditions across a finite strip the row to row transfer matrix is diagonalized by the commuting transfer matrices method from the exact solution we obtain an independent derivation of the interfacial tension of the six vertex model in the anti ferroelectric phase the nature of the corresponding integrable boundary condition on the xxz spin chain is also discussed

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

coupling an eulerian fluid calculation to a lagrangian solid calculation with the ghost fluid method we propose a numerical method for modeling multimaterial flows where the domain is decomposed into separate eulerian and lagrangian subdomains that is the equations are written in eulerian form in one subdomain and in lagrangian form in the other subdomain this is of interest for example when considering the effect of underwater explosions on the hull of a ship or the impact of a low speed projectile on a soft explosive target on the one hand high speed fluid flows are traditionally modeled by applying shock capturing schemes to the compressible euler equations to avoid problems associated with tangling of a lagrangian mesh on the other hand solid dynamics calculations are traditionally carried out using lagrangian numerical methods to avoid problems associated with numerical smearing in eulerian calculations we use the ghost fluid method to create accurate discretizations across the eulerian lagrangian interface the numerical method is presented in both one and two spatial dimensions three dimensional extensions to the interface coupling method are straightforward

on the vectoral ekeland s variational principle and minimal points in product spaces abstract phelps 13 noticed that the scalar ekeland s variational principle evp is equivalent to the existence of a minimal point of the epigraph of the corresponding function with respect to an appropriate order attouch and riahi 1 showed that evp is equivalent to the existence of maximal points with respect to cones satisfying some additional conditions taking these into account gopfert and tammer 6 7 established a maximal point theorem in a product space the aim of this paper is to obtain several minimal point theorems in product spaces and the corresponding variants of the vectorial evp

ekeland s variational principle and its equivalent theorems in vector optimization we present a simple proof of vectorial ekeland s variational principle vectorial caristi s fixed point theorem and vectorial takahashi s nonconvex minimization theorem equivalences between the three theorems are also given

a pre order principle and set valued ekeland variational principle abstract we establish a pre order principle from the principle we obtain a very general set valued ekeland variational principle where the objective function is a set valued map taking values in a quasi ordered linear space and the perturbation contains a family of set valued maps satisfying certain property from this general set valued ekeland variational principle we deduce a number of particular versions of set valued ekeland variational principle which include many known ekeland variational principles their improvements and some new results

exact solution of the six vertex model with domain wall boundary conditions disordered phase the six vertex model or the square ice model with domain wall boundary conditions dwbc has been introduced and solved for finite n by korepin and izergin the solution is based on the yang baxter equations and it represents the free energy in terms of an n n hankel determinant paul zinn justin observed that the izergin korepin formula can be re expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction we use this observation to obtain the large n asymptotics of the six vertex model with dwbc in the disordered phase the solution is based on the riemann hilbert approach and the deift zhou nonlinear steepest descent method as was noticed by kuperberg the problem of enumeration of alternating sign matrices the asm problem is a special case of the six vertex model we compare the obtained exact solution of the six vertex model with known exact results for the 1 2 and 3 enumerations of asms and also with the exact solution on the so called free fermion line we prove the conjecture of zinn justin that the partition function of the six vertex model with dwbc has the asymptotics z n sim cn kappa e n 2 f as n and we find the exact value of the exponent

distribution of the first particle in discrete orthogonal polynomial ensembles we show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure if the difference or q log derivative of the weight function is rational in a number of classical special cases the recurrence procedure is equivalent to the difference and q painleve equations of 10 17 our approach is based on the formalism of discrete integrable operators and discrete riemann hilbert problems developed in 3 4

alternating direction ghost fluid methods for solving the heat equation with interfaces abstract this work presents two new alternating direction implicit adi schemes for solving parabolic interface problems in two and three dimensions first the ghost fluid method gfm is adopted for the first time in the literature to treat interface jumps in the adi framework which results in symmetric and tridiagonal finite difference matrices in each adi step the proposed gfm adi scheme achieves a first order of accuracy in both space and time as confirmed by numerical experiments involving complex interface shapes and spatial temporal dependent jumps the gfm adi scheme also maintains the adi efficiency the computational complexity for each time step scales as o n for a total degree of freedom n in higher dimensions second a new matched interface and boundary mib scheme is constructed which downgrades the quadratic polynomial bases in the existing second order mib to linear ones interestingly the resulting mib adi or madi scheme produces the same finite difference matrices as the gfm adi scheme in all dimensions hence the madi scheme can be regarded as an improvement of the gfm because it calculates tangential jumps which are omitted in the gfm consequently the present madi scheme is constantly more accurate than the gfm adi in all numerical examples while keeping the same computational efficiency nevertheless the madi scheme is semi implicit due to tangential jump approximations while the gfm adi scheme is fully implicit without tangential corrections thus the gfm adi scheme could be more stable than the madi scheme when a huge contrast is presented in diffusion coefficients

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

emptiness formation probability of the six vertex model and the sixth painleve equation we show that the emptiness formation probability of the six vertex model with domain wall boundary conditions at its free fermion point is a tau function of the sixth painleve equation using this fact we derive asymptotics of the emptiness formation probability in the thermodynamic limit

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

two dimensional yang mills theory painleve equations and the six vertex model we show that the chiral partition function of two dimensional yang mills theory on the sphere can be mapped to the partition function of the homogeneous six vertex model with domain wall boundary conditions in the ferroelectric phase a discrete matrix model description in both cases is given by the meixner ensemble leading to a representation in terms of a stochastic growth model we show that the partition function is a particular case of the z measure on the set of young diagrams yielding a unitary matrix model for chiral yang mills theory on s2 and the identification of the partition function as a tau function of the painlev v equation we describe the role played by generalized non chiral yang mills theory on s2 in relating the meixner matrix model to the toda chain hierarchy encompassing the integrability of the six vertex model we also argue that the thermodynamic behaviour of the six vertex model in the disordered and antiferroelectric phases are captured by particular q deformations of two dimensional yang mills theory on the sphere

accuracies and conservation errors of various ghost fluid methods for multi medium riemann problem since the original ghost fluid method ogfm was proposed by fedkiw et al in 1999 5 a series of other gfm based methods such as the gas water version gfm gwgfm the modified gfm mgfm and the real gfm rgfm have been developed subsequently systematic analysis however has yet to be carried out for the various gfms on their accuracies and conservation errors in this paper we develop a technique to rigorously analyze the accuracies and conservation errors of these different gfms when applied to the multi medium riemann problem with a general equation of state eos by analyzing and comparing the interfacial state provided by each gfm to the exact one of the original multi medium riemann problem we show that the accuracy of interfacial treatment can achieve third order accuracy in the sense of comparing to the exact solution of the original mutli medium riemann problem for the mgfm and the rgfm while it is of at most first order accuracy for the ogfm and the gwgfm when the interface approach is actually near in balance similar conclusions are also obtained in association with the local conservation errors a special test method is exploited to validate these theoretical conclusions from the numerical viewpoint

a ghost fluid method for large eddy simulations of premixed combustion in complex geometries in this paper a new ghost fluid method for interfaces of finite thickness is described it allows to compute efficiently turbulent premixed flames with a finite thickness in low mach flows a level set algorithm is used to track accurately the flame and to define the overlapping region where the burned and unburned gases satisfy the jump conditions these algorithms are combined with a fractional step method to alleviate the acoustic cfl constraint the full algorithm is verified for simple flame vortex interactions and it is validated by computing a turbulent flame anchored by a triangular flame holder finally the algorithm is applied in the les of an industrial lean premixed swirl burner

optimal error estimates for linear parabolic problems with discontinuous coefficients a finite element discretization is proposed and analyzed for a linear parabolic problems with discontinuous coefficients due to low global regularity of the solution it seems difficult to achieve optimal order of convergence with classical finite element methods numer math 79 1998 pp 175 202 in this paper we have used a finite element discretization where interface triangles are assumed to be curved triangles instead of straight triangles as in classical finite element methods optimal order error estimates in l2 and h1 norms are shown to hold even if the regularity of the solution is low on the whole domain while the continuous time galerkin method is discussed for the spatially discrete scheme the discontinuous galerkin method is analyzed for the fully discrete scheme the interfaces and boundaries of the domains are assumed to be smooth for our purpose

ekelandʼs variational principle for an l 0 valued function on a complete random metric space abstract motivated by the recent work on conditional risk measures this paper studies the ekeland s variational principle for a proper lower semicontinuous and lower bounded l 0 valued function where l 0 is the set of equivalence classes of extended real valued random variables on a probability space first we prove a general form of ekeland s variational principle for such a function defined on a complete random metric space then we give a more precise form of ekeland s variational principle for such a local function on a complete random normed module finally as applications we establish the bishop phelps theorem in a complete random normed module under the framework of random conjugate spaces

convergence of the ghost fluid method for elliptic equations with interfaces this paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps a weak formulation of the problem is first presented which then yields the existence and uniqueness of a solution to the problem by classical methods it is shown that the application of the ghost fluid method by fedkiw kang and liu to this problem can be obtained in a natural way through discretization of the weak formulation an abstract framework is given for proving the convergence of finite difference methods derived from a weak problem and as a consequence the ghost fluid method is proved to be convergent

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

functional relations in nineteen vertex models with domain wall boundaries this work is concerned with functional properties shared by partition functions of nineteen vertex models with domain wall boundary conditions in particular we describe both izergin korepin and fateev zamolodchikov models with the aforementioned boundary conditions and show that their partition functions are governed by a system of functional equations originating from the associated yang baxter algebra

fixed point theorems for multivalued mappings on complete metric spaces on obtient une version multivaluee du theoreme de point fixe de caristi qui est equivalent au principe e variationnel du a ekeland on utilise ce resultat pour obtenir un theoreme qui deduit le theoreme de nadler et un theoreme de point fixe commun pour une application a 1 valeur et une application multivaluee

the ghost fluid method for compressible gas water simulation an analysis is carried out for the ghost fluid method gfm based algorithm as applied to the gas water riemann problems which can be construed as two single medium gfm riemann problems it is found that the inability to provide correct and consistent riemann waves in the respective real fluids by these two gfm riemann problems may lead to inaccurate numerical results based on this finding two conditions are suggested and imposed for the ghost fluid status in order to ensure that correct and consistent riemann waves are provided in the respective real fluids during the numerical decomposition of the singularity using these two conditions to analyse some of the existing gfm based algorithms such as the original gfm j comput phys 152 1999 457 the new version gfm j comput phys 166 2001 1 j comput phys 175 2002 200 and the modified gfm mgfm j comput phys 190 2003 651 it is found that there are ranges of conditions for each type of solution where either the original gfm or the new version gfm or both are unable to provide correct or consistent riemann waves in one of the real fluids within these ranges examples can be found such that either the original gfm or the new version gfm or both are unable to provide accurate results the mgfm is also found to encounter difficulties when applied to nearly cavitating flow various examples are presented to demonstrate the conclusions obtained the mgfm with proposed modification when applied to nearly cavitating flow is then found to be quite robust and can provide relatively reasonable results

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

a dimension by dimension splitting immersed interface method for heat conduction equation with interfaces the numerical method proposed in this paper is an improvement of the adi method by li and mayo 1994 the proposed method is unconditionally stable for both two and three dimensional heat conduction interface problems while li s adi method is only stable for two dimensional problems the method is a modification of a locally one dimensional lod difference scheme with correction term added to the right hand side of the standard lod difference scheme at irregular points the correction term is determined so that the local truncation error is of order o h at irregular points then the method is two order convergent in both time and space directions numerical examples show good agreement with exact solutions and confirm the order of convergence and stability

domino tilings and the six vertex model at its free fermion point at the free fermion point the six vertex model with domain wall boundary conditions dwbc can be related to the aztec diamond a domino tiling problem we study the mapping on the level of complete statistics for general domains and boundary conditions this is obtained by associating with both models a set of non intersecting lines in the lindstrom gessel viennot lgv scheme one of the consequences for dwbc is that the boundaries of the ordered phases are described by the airy process in the thermodynamic limit

the immersed interface technique for parabolic problems with mixed boundary conditions a finite difference scheme is presented for a parabolic problem with mixed boundary conditions we use an immersed interface technique to discretize the neumann condition and we use the shortley weller approximation for the dirichlet condition the proof of a discrete maximum principle is given as well as the proof of convergence of the scheme this convergence is also validated on numerical examples

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

nonintersecting brownian motions on the unit circle we consider an ensemble of n nonintersecting brownian particles on the unit circle with diffusion parameter n 1 2 which are conditioned to begin at the same point and to return to that point after time t but otherwise not to intersect there is a critical value of t which separates the subcritical case in which it is vanishingly unlikely that the particles wrap around the circle and the supercritical case in which particles may wrap around the circle in this paper we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution we also give a streamlined approach to identifying the pearcey and tacnode processes in scaling limits the formula of the tacnode correlation kernel is new and involves a solution to a lax system for the painleve ii equation of size 2 2 the proofs are based on the determinantal structure of the ensemble asymptotic results for the related system of discrete gaussian orthogonal polynomials and a formulation of the correlation kernel in terms of a double contour integral

exact solution of the six vertex model with domain wall boundary conditions critical line between ferroelectric and disordered phases this is a continuation of the papers of bleher and fokin commun math phys 268 223 284 2006 and of bleher and liechty commun math phys 286 777 801 2009 in which the large n asymptotics is obtained for the partition function z n of the six vertex model with domain wall boundary conditions in the disordered and ferroelectric phases respectively in the present paper we obtain the large n asymptotics of z n on the critical line between these two phases

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

six vertex model with domain wall boundary conditions and one matrix model the partition function of the six vertex model on a square lattice with domain wall boundary conditions dwbc is rewritten as a hermitian one matrix model or a discretized version of it similar to sums over young diagrams depending on the phase the expression is exact for finite lattice size which is equal to the size of the corresponding matrix in the thermodynamic limit the matrix integral is computed using traditional matrix model techniques thus providing a complete treatment of the bulk free energy of the six vertex model with dwbc in the different phases in particular in the antiferroelectric phase the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions

thermodynamic limit of the six vertex model with domain wall boundary conditions we address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model here we compare the bulk free energy for periodic and domain wall boundary conditions using a determinant representation for the partition function with domain wall boundary conditions we derive toda differential equations and solve them asymptotically in order to extract the bulk free energy we find that it is different and bears no simple relation to the free energy for periodic boundary conditions the six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics alternating sign matrices this implies new results for the weighted counting for large size alternating sign matrices finally we comment on the interpretation of our results in particular in connection with domino tilings dimers on a square lattice

a ghost fluid level set methodology for simulating multiphase electrohydrodynamic flows with application to liquid fuel injection in this paper we present the development of a sharp numerical scheme for multiphase electrohydrodynamic ehd flows for a high electric reynolds number regime the electric potential poisson equation contains ehd interface boundary conditions which are implemented using the ghost fluid method gfm the gfm is also used to solve the pressure poisson equation the methods detailed here are integrated with state of the art interface transport techniques and coupled to a robust high order fully conservative finite difference navier stokes solver test cases with exact or approximate analytic solutions are used to assess the robustness and accuracy of the ehd numerical scheme the method is then applied to simulate a charged liquid kerosene jet

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19

an adaptive ghost fluid finite volume method for compressible gas water simulations an adaptive ghost fluid finite volume method is developed for one and two dimensional compressible multi medium flows in this work it couples the real ghost fluid method gfm c w wang t g liu b c khoo a real ghost fluid method for the simulation of multi medium compressible flow siam j sci comput 28 2006 278 302 and the adaptive moving mesh method h z tang t tang moving mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 h z tang t tang p w zhang an adaptive mesh redistribution method for non linear hamilton jacobi equations in two and three dimensions j comput phys 188 2003 543 572 and thus combines their advantages this work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the gfm around the material interface and other discontinuities besides the improvement of flow field resolution the adaptive gfm also largely increases the computational efficiency several numerical experiments are conducted to demonstrate robustness and efficiency of the current method they include several 1d and 2d gas water flow problems involving a large density gradient at the material interface and strong shock interface interactions the results show that our algorithm can capture the shock waves and the material interface accurately and is stable and robust even for solutions with large density and pressure gradients

exact solution of the six vertex model with domain wall boundary conditions antiferroelectric phase we obtain the large n asymptotics of the partition function zn of the six vertex model with domain wall boundary conditions in the antiferroelectric phase region with the weights a sinh t b sinh t c sinh 2 t we prove the conjecture of zinn justin that as n zn c 4 n f 1 o n 1 where and f are given by explicit expressions in and t and 4 z is the jacobi theta function the proof is based on the riemann hilbert approach to the large n asymptotic expansion of the underlying discrete orthogonal polynomials and on the deift zhou nonlinear steepest descent method 2009 wiley periodicals inc

determinant formula for the six vertex model the partition function of a six vertex model with domain wall boundary conditions is considered on the finite lattice the authors show that the partition function satisfies a recursive relation they solve the recursion relation by a determinant formula this gives a determinant representation for the partition function they use the quantum inverse scattering method qism