structure of certain chebyshev type polynomials in onsager s algebra representation in this report we present a systematic account of mathematical structures of certain special polynomials arisen from the energy study of the superintegrable n state chiral potts model with a finite number of sizes the polynomials of low lying sectors are represented in two different forms one of which is directly related to the energy description of superintegrable chiral potts z n spin chain via the representation theory of onsager s algebra both two types of polynomials satisfy some n 1 term recurrence relations and nth order differential equations polynomials of one kind reveal certain chebyshev like properties here we provide a rigorous mathematical argument for cases n 2 3 and further raise some mathematical conjectures on those special polynomials for a general n

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

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

the three dimensional ising model revisited numerically several basic universal amplitude ratios are studied afresh for three dimensional nearest neighbor ising models in revising earlier work modern estimates of the critical temperature and exponents are used in conjunction with biased inhomogeneous differential approximants to extrapolate the longest available series expansions to find the critical amplitudes c for the susceptibility 1 for the correlation length 1 a for the specific heat c t and b for the spontaneous magnetization m0 we find c c 4 95 0 15 1 1 1 960 0 01 a a 0 523 0 009 a c b 0 0581 0 0010 while a 1 3 verifies hyperuniversality to within 0 8 a method for calculating amplitude ratios which allows for corrections to scaling yields estimates for c c and 1 1 in excellent agreement with those derived from the individual amplitudes finally explicit formulae are given for the numerical evaluation of t 1 t c t and m0 t over the full temperature range from criticality to t 0 and corresponding plots and convenient near critical representations are also presented

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

a two grid finite difference scheme for nonlinear parabolic equations we present a two level finite difference scheme for the approximation of nonlinear parabolic equations discrete inner products and the lowest order raviart thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme analysis of the scheme is given assuming an implicit time discretization in this two level scheme the full nonlinear problem is solved on a coarse grid of size h the nonlinearities are expanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points the resulting linear but nonsymmetric system is solved on a fine grid of size h some a priori error estimates are derived which show that the discrete l infty l2 and l2 h1 errors are o h 2 h 4 d 2 delta t where d geq 1 is the spatial dimension

a two grid block centered finite difference method for darcy forchheimer flow in porous media a two grid block centered finite difference method is proposed for solving the two dimensional darcy forchheimer model describing non darcy flow in porous media to construct the two grid method we modify the original nonlinear elliptic operator of darcy forchheimer flow to a twice continuously differentiable one by introducing a small and positive parameter varepsilon by using the two grid method solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid optimal order error estimates for pressure and velocity in discrete l 2 norms are obtained some numerical examples are given to show the accuracy and efficiency of the presented method

error analysis of the semi discrete local discontinuous galerkin method for compressible miscible displacement problem in porous media in this paper local discontinuous galerkin method for flow and transport is introduced for the coupled system of compressible miscible displacement problem optimal error estimates in l 0 t l 2 norm not only for the solution itself but also for the auxiliary variables are derived the main technical difficulties in the analysis include the nonlinearity the coupling of the models and the treatment of the inter element jump terms which arise from the discontinuous nature of the numerical method the numerical results illustrate the accuracy and capability of the method

a parallel cgs block centered finite difference method for a nonlinear time fractional parabolic equation abstract in this article a parallel conjugate gradient squared cgs block centered finite difference scheme is introduced and analyzed to cast about the numerical solution of a nonlinear time fractional parabolic equation with the neumann condition and a nonlinear reaction term the unconditionally stable result which just depends on initial value and source item is derived some a priori estimates of discrete l 2 norm with optimal order of convergence o t 2 h 2 k 2 with pressure and velocity are established on both uniform and non uniform rectangular grids where t is the time step h and k are maximal mesh sizes of x and y directional grids in our model because the simulation is iterated over a series of time steps it is most beneficial if all the calculation units in a time step can be run separately in cgs algorithm adding openmp instructions to the circulation calculations in an iterative step can realize parallel computing to examine the efficiency and accuracy of the proposed method numerical experiments using the schemes are studied the results clearly show the benefit of using the proposed approach in terms of execution time reduction and speedup with respect to the sequential running in a single thread

the large n limits of the chiral potts model abstract in this paper we study the large n limits of the integrable n state chiral potts model three chiral solutions of the star triangle equations are derived with states taken from all integers or from a finite or infinite real interval these solutions are expected to be chiral field lattice deformations of parafermionic conformal field theories a new two sided hypergeometric identity is derived as a corollary

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

phase diagram in the generalized chiral clock models abstract using the results of the integrable chiral potts model we speculate about the possible phase diagram of the generalized chiral clock model in the integrable model it is found that the chiral fields are almost proportional to one another here we consider a chiral model with all the chiral fields exactly proportional to one another with fixed proportionality constants given by the integrable case this reduces the chiral field variables to just one denoted by the model thus defined includes the generalized clock model or z n model at 0 or 1 2 and the chiral clock models as special cases using the results of many authors on the z n model in the so called symmetric case with vertical and horizontal interactions equal and the behavior of the system at t 0 we can get a pretty good picture of the phase diagram of this chiral model the exact results show that the commonly held belief that the massless phase in the z n model for n 5 is in the same universality class as the classical xy model and the belief that the commensurate incommensurate phase transition is of the pokrovskii talapov type and the incommensurate fluid transition is of kosterlitz thouless type may not be correct as universality may not even hold on these phase boundaries

superconvergence for rectangular mixed finite elements in this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems for the rectangular finite elements of raviart and thomas 19 and for those of brezzi et al 4 we prove that the distance inl 2 between the approximate solution and a projection of the exact one is of higher order than the error itself this result is exploited to obtain superconvergence at gaussian points and to construct higher order approximations by a local postprocessing

solvable models in statistical mechanics from onsager onward there is now a whole field in mathematical physics concerned with solvable models in statistical mechanics field theory and related areas we indicate the influence that onsager s solution of the planar ising model has had and continues to have on this field

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

on zamolodchikov s solution of the tetrahedron equations the tetrahedron equations arise in field theory as the condition for thes matrix in 2 1 dimensions to be factorizable and in statistical mechanics as the condition that the transfer matrices of three dimensional models commute zamolodchikov has proposed what appear from numerical evidence and special cases to be non trivial particular solutions of these equations but has not fully verified them here it is proved that they are indeed solutions

a combined mixed and discontinuous galerkin method for compressible miscible displacement problem in porous media in this paper we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media the flow equation is solved by the mixed finite element method and the transport equation is approximated by a discontinuous galerkin method the scheme is continuous in time and a priori hp error estimates is presented

n abelian quotient categories abstract let c be an n 2 angulated category with an n suspension functor n and x be a cluster tilting subcategory of c then we show that the quotient category c x is an n abelian category and c x is equivalent to an n cluster tilting subcategory of an abelian category mod n x in addition if c has a serre functor we also prove that mod n x is gorenstein of gorenstein dimension at most n as an application we generalize recent results of jacobsen jorgensen and koenig zhu

free energy of the chiral potts model in the scaling region we explicitly calculate the free energy of the general solvablen state chiral potts model in the scaling region fort t c we do this from both of the two available results for the free energy and verify that they are mutually consistent ift t c t then we find that c t has a taylor expansion in powers oft 2 n together with higher order non scaling terms of ordert ort logt

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

the q operator for root of unity symmetry in the six vertex model we construct the explicit q operator incorporated with the sl2 loop algebra symmetry of the six vertex model at roots of unity the functional relations involving the q operator the six vertex transfer matrix and fusion matrices are derived from the bethe equation parallel to the onsager algebra symmetry discussion in the superintegrable n state chiral potts model we show that the whole set of functional equations is valid for the q operator direct calculations in certain cases are also given here for clearer illustration about the nature of the q operator in the symmetry study of root of unity six vertex model from the functional relation aspect

characteristic block centered finite difference method for compressible miscible displacement in porous media in this paper the characteristic block centered finite difference method is introduced and analyzed to solve compressible miscible displacement in porous media error estimates for the pressure velocity concentration and its flux in different discrete norms are established rigorously and carefully on non uniform grids finally some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis

a block centered finite difference method for the darcy forchheimer model a block centered finite difference scheme is introduced to solve the nonlinear darcy forchheimer equation in which the velocity and pressure can be approximated simultaneously the second order error estimates for both pressure and velocity are established on a nonuniform rectangular grid numerical experiments using the scheme show the consistency of the convergence rates of our method with the theoretical analysis

baxter equations and the zamolodchikov model abstract a summary is given of the significance of the star triangle or yang baxter equations in the solution of two dimensional statistical mechanical models the extension to the three dimensional tetrahedron relations is quoted and it is shown how this can be used to obtain the partition function of the zamolodchikov model for an n lattice the n result differs from that recently obtained by bazhanov and stroganov

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

non isothermal compressible gas flow for the simulation of an enhanced gas recovery application in this work we present a framework for numerical modeling of co 2 injection into porous media for enhanced gas recovery egr from depleted reservoirs physically we have to deal with non isothermal compressible gas flows resulting in a system of coupled non linear pdes we describe the mathematical framework for the underlying balance equations as well as the equations of state for mixing gases we use an object oriented finite element method implemented in c the numerical model has been tested against an analytical solution for a simplified problem and then applied to co 2 injection into a real reservoir numerical modeling allows to investigate physical phenomena and to predict reservoir pressures as well as temperatures depending on injection scenarios and is therefore a useful tool for applied numerical analysis

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

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

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

the baxter bazhanov stroganov model separation of variables and the baxter equation the baxter bazhanov stroganov model also known as the 2 model has attracted much interest because it provides a tool for solving the integrable chiral potts model it can be formulated as a face spin model or via cyclic l operators using the latter formulation and the sklyanin kharchev lebedev approach we give the explicit derivation of the eigenvectors of the component bn of the monodromy matrix for the fully inhomogeneous chain of finite length for the periodic chain we obtain the baxter t q equations via separation of variables the functional relations for the transfer matrices of the 2 model guarantee nontrivial solutions to the baxter equations for the n 2 case which is the free fermion point of a generalized ising model the baxter equations are solved explicitly

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

the chiral potts models revisited we review some exact results obtained so far in the chiral potts models and translate these results into language more transparent to physicists so that experts in monte carlo calculations high and low temperature expansions and various other methods can use them we pay special attention to the interfacial tension r between thek state and thek r state by examining the ground states it is seen that the integrable line ends at a superwetting point on which the relation r r 1 is satisfied so that it is energetically neutral to have one interface or more we present also some partial results on the meaning of the integrable line for low temperatures where it lives in the nonwet regime we make baxter s exact results more explicit for the symmetric case by performing a bethe ansatz calculation with open boundary conditions we confirm a dilogarithm identity for the low temperature expansion which may be new we propose a new model for numerical studies this model has only two variables and exhibits commensurate and incommensurate phase transitions and wetting transitions near zero temperature it appears to be not integrable except at one point and at each temperature there is a point where it is almost identical with the integrable chiral potts model

numerical methods for a model for compressible miscible displacement in porous media a nonlinear parabolic system is derived to describe compressible miscible displace ment in a porous medium the system is consistent with the usual model for incompressible miscible displacement two finite element procedures are introduced to approximate the concentration of one of the fluids and the pressure of the mixture the concentration is treated by a galerkin method in both procedures while the pressure is treated by either a lalerkin method or by a parabolic mixed finite element method optimal order estimates in l2 and essentially optimal order estimates in lx are derived for the errors in the approximate solutions for both methods introduction we shall consider the single phase miscible displacement of one compressible fluid by another in a porous medium under the assumptions that no volume change results from the mixing of the components and that a pressure den sity relation exists for each component in a form that is independent of the mixing these equations of state will imply that the fluids are in the liquid state our model will represent a direct generalization of the model 3 4 7 that has been treated extensively for incompressible miscible displacement the reservoir s will be taken to be of unit thickness and will be identified with a bounded domain in r2 we shall omit gravitational terms for simplicity of exposi tion no significant mathematical questions arise when the lower order terms are included

simple ising models still thrive a review of some recent progress abstract a selection of recent work by various authors concerning spin 1 2 ising modls with pair couplings is reviewed topics addressed include the universality of corrections to scaling and nonlinear scaling fields in planar ising models aspects of many particle correlation functions the testing and verification of scaling hypotheses for local magnetizations in the presence of walls and boundaries including the singular nature of the critical region scaling functions and the decay of the influence of a far wall a soluble model of a surface roughening transition nonuniversal critical effects of one dimensional perturbations in planar ising models the effects of spatially anisotropic further neighbor interactions in particular annni models in d 2 dimensions and soluble mock annni models in 2 and 3 dimensions

interfacial tension of the chiral potts model we calculate the interfacial tension of then state chiral potts model by solving the functional relations for the transfer matrices of the model with skewed boundary conditions our result is valid for the general physical model with positive boltzmann weights and at all subcritical temperatures the interfacial tension has been calculated previously for the superintegrable chiral potts model with skewed boundary conditions usingz invariance baxter has argued that the interfacial tension of this model should be the same as the interfacial tension of the general physical model we show that this is indeed the case

mixed finite element methods for compressible miscible displacement in porous media a differential system describing compressible miscible displacement in a porous medium is given the concentration equation is treated by a galerkin method and the pressure equation is treated by a parabolic mixed fi nite element method optimal order estimates in l and almost optimal order estimates in l are obtained for the errors in the approximate solutions under the condition that h logh 2 0 this condition is much weaker than one given earlier by douglas and roberts for the same model furthermore we obtain the l l i2 estimates for the time derivatives of the concentration and the pressure which were not given by the above authors in addition we also consider newer mixed spaces in two or three dimensions

csos models descending from chiral potts models degeneracy of the eigenspace and loop algebra monodromy matrices of the model are known to satisfy a yang baxter equation with a six vertex r matrix as the intertwiner the commutation relations of the elements of the monodromy matrices are completely determined by this r matrix we show the reason why in the superintegrable case the eigenspace is degenerate but not in the general case we then show that the eigenspaces of special csos models descending from the chiral potts model are also degenerate the existence of an quantum loop algebra or subalgebra in these models is established by showing that the serre relations hold for the generators the highest weight polynomial or the drinfeld polynomial of the representation is obtained by using the method of baxter for the superintegrable case as a byproduct the eigenvalues of all such csos models are given explicitly

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

on a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media we study a model describing a compressible and miscible displacement in a porous medium it consists of a coupled system of nonlinear parabolic partial differential equations using nonclassical estimates and renormalization tools we prove the existence of relevant weak solutions for the problem this is the first existence result obtained for a transport model containing both the coupling due to the compressibility assumption and the coupling due to the concentration dependent viscosity

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

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

tetrahedron equations and the relativistic s matrix of straight strings in 2 1 dimensions the quantums matrix theory of straight strings infinite one dimensional objects like straight domain walls in 2 1 dimensions is considered thes matrix is supposed to be purely elastic and factorized the tetrahedron equations which are the factorization conditions are investigated for the special two colour model the relativistic three strings matrix which apparently satisfies this tetrahedron equation is proposed

finite difference methods for two phase incompressible flow in porous media two phase incompressible flow in porous media is governed by a system of nonlinear partial differential equations convection physically dominates diffusion and the object of this paper is to develop a finite difference procedure that reflects this dominance the pressure equation which is elliptic in appearance is discretized by a standard five point difference method the concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms a convergence analysis is given for the method

bethe ansatz type equations for the fateev zamolodchikov spin model the eigenvalues of the fateev zamolodchikov zn invariant model transfer matrix are found for n odd their zeros in the complex plane of the rapidity variable are shown to satisfy a set of bethe ansatz type equations similar to those obtained for the integrable xxz chains the eigenvalue for a filled sea of n 1 strings gives the free energy found by the matrix inversion method

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

on solving the ising model functional relation without elliptic functions the exactly solvable n state chiral potts model differs from previous solvable lattice models in that its mathematics leads to hyperelliptic rather than ordinary elliptic functions it is still an open question whether these hyperelliptic functions provide a useful computational tool in particular it has recently been shown that the functional relation for the eigenvalues of the transfer matrix can be solved directly without introducing such functions when n 2 the model reduces to the ising model and this method provides a novel and straightforward way of solving the ising model in fact the method simplifies considerably here we present the resulting technique

d abelian quotients of d 2 angulated categories abstract let t be a triangulated category if t is a cluster tilting object and i add t is the ideal of morphisms factoring through an object of add t then the quotient category t i is abelian this is an important result of cluster theory due to keller reiten and konig zhu more general conditions which imply that t i is abelian were determined by grimeland and the first author now let t be a suitable d 2 angulated category for an integer d 1 if t is a cluster tilting object in the sense of oppermann thomas and i add t is the ideal of morphisms factoring through an object of add t then we show that t i is d abelian the notions of d 2 angulated and d abelian categories are due to geiss keller oppermann and jasso they are higher homological generalisations of triangulated and abelian categories which are recovered in the special case d 1 we actually show that if end t t is the endomorphism algebra of t then t i is equivalent to a d cluster tilting subcategory of mod in the sense of iyama this implies that t i is d abelian moreover we show that is a d gorenstein algebra more general conditions which imply that t i is d abelian will also be determined generalising the triangulated results of grimeland and the first author

on the corrections to scaling in three dimensional ising models the leading correction to scaling amplitudes for the spin 1 2 nearest neighbor sc bcc and fee ising models are considered with the particular aim of determining their signs on the basis of previous two variable series analyses by chen fisher and nickel and renormalization group 4 d expansions it is concluded that the correction amplitudes for the susceptibility correlation length specific heat and spontaneous magnetization arenegative for all three lattices thus for example the effective exponent eff t asymptotically approaches the true susceptibility exponent fromabove other earlier and more recent high temperature series and field theoretic analyses are seen to be consistent with this result however the usual nonasymptotic perturbative field theoretic approaches are essentially committed to positive correction amplitudes the question of the signs therefore relates directly to the applicability of these non asymptotic field theoretic calculations to three dimensional ising models as well as to different experimental systems

n angulated categories we dene n angulated categories by modifying the axioms of triangulated categories in a natural way we show that heller s parametrization of pre triangulations extends to pre n angulations we obtain a large class of examples of n angulated cate gories by considering n 2 cluster tilting subcategories of triangulated categories which are stable under the n 2 nd power of the suspension functor as an application we show how n angulated calabi yau categories yield triangulated calabi yau categories of higher calabi yau dimension finally we sketch a link to algebraic geometry and string theory

a two grid block centered finite difference method for nonlinear non fickian flow model in this paper a two grid block centered finite difference scheme is introduced and analyzed to solve the nonlinear parabolic integro differential equation arising in modeling non fickian flow in porous media this method is considered where the nonlinear problem is solved only on a coarse grid of size h and a linear problem is solved on a fine grid of size h error estimates are established on non uniform rectangular grid which show that the discrete l l2 and l2 h1 errors are o t h 2 h 3 finally some numerical experiments are presented to show the efficiency of the two grid method and verify that the convergence rates are in agreement with the theoretical analysis