solutions of the yang baxter equation for isotropic quantum spin chains the author considers solutions of the yang baxter equation such that the logarithmic derivative of the transfer matrix yields a quantum spin hamiltonian which is isotropic in spin space i e su 2 invariant four such solutions are known for each value of the spin s for s 1 2 they degenerate into the same solution and for s 1 they only give three different solutions for s or 6 he shows that these are the only solutions which are su 2 invariant except for s 3 when there is a fifth solution

algebraic bethe ansatz for the one dimensional bariev chain the algebraic bethe ansatz is formulated for the one dimensional bariev chain as a consequence the eigenvectors eigenvalues and bariev s bethe equations are rederived in a systematic algebraic way

constructing homogeneous gorenstein ideals abstract in 1983 kustin and miller introduced a construction of gorenstein ideals in local gorenstein rings starting from smaller such ideals we review and modify their construction in the case of graded rings and discuss it within the framework of gorenstein liaison theory we determine invariants of the constructed ideal concerning the problem of when a given gorenstein ideal can be obtained by the construction we derive a necessary condition and exhibit a gorenstein ideal that cannot be obtained using the construction

tight double linkage of gorenstein algebras abstract a gorenstein ideal k in a local ring r is in the class if there is a sequence of linked ideals i 0 i 1 i 2 n k with i 0 a complete intersection and i 2 i gorenstein for all i if k is in then k is linked to a complete intersection by a sequence of tight double links this means there is a sequence i 0 j 0 j 1 j 2m k such that for each p j 2 p 1 is an almost complete intersection of the form b y w and its immediate neighbors are linked to it by the similar regular sequences b y and b w respectively in this sense j 2 p and j 2 p 2 are tightly linked gorenstein ideals let k be an algebraically closed field and a k x 1 x q k with k in the k algebra a is rigid if and only if it is linked to a regular complete intersection by a sequence of semi generic tight double links it follows that if a is rigid then it is regular in codimension six

maximally extended sl 2 2 as a quantum double we derive the universal r matrix of the quantum deformed enveloping algebra of centrally extended using drinfeld s quantum double construction we are led to enlarging the algebra by additional generators corresponding to an automorphism for this maximally extended algebra we construct a consistent hopf algebra structure where the extensions exhibit several uncommon features we determine the corresponding universal r matrix containing some non standard functions curiously this hopf algebra has one extra deformation parameter for which the r matrix does not factorize into products of exponentials

quantum integrability in systems with finite number of levels we consider the problem of defining quantum integrability in systems with finite number of energy levels starting from commuting matrices and construct new general classes of such matrix models with a given number of commuting partners we argue that if the matrices depend on a real parameter one can define quantum integrability from this feature alone leading to specific results such as exact solvability poissonian energy level statistics and to level crossings

eigenvectors of the multi component generalization of the six vertex model abstract the most general multi component solution of the star triangle equations under an ice type restriction is considered the transfer matrix of this model is diagonalized and the eigenvectors are written down explicitly using the quantum inverse scattering formalism the eigenvectors are shown to have simple symmetry properties under an exchange of arguments

4 symmetry of the transfer matrix for the one dimensional hubbard model the so 4 invariance of the transfer matrix for the one dimensional hubbard model is clarified from the viewpoint of the quantum inverse scattering method we demonstrate the so 4 symmetry by means of the fermionic l operator and the fermionic r matrix which satisfy the graded yang baxter relation the transformation law of the fermionic l operator under the so 4 rotation is identified with a kind of gauge transformation which determines the corresponding transformation of the fermionic creation and annihilation operators under the so 4 rotation the transfer matrix is confirmed to be invariant under the so 4 rotation which ensures the so 4 invariance of the conserved currents including the hamiltonian furthermore we show that the representation of the higher conserved currents in terms of the clifford algebra gives manifestly so 4 invariant forms

linkage and sums of ideals it is shown under mild conditions that the sum of transversal ideals in a regular local ring cannot lie in the linkage class of a complete intersection for a sum of geometrically linked cohen macaulay ideals we compute the depths of the conormal module arld the first koszul homology as applications we construct general examples of ideals which are strongly cohenmacaulay strongly nonobstructed but not in the linkage class of a complete intersection and gorenstein ideals which are strongly nonobstructed but not syzygetic

structure theory for a class of grade four gorenstein ideals asstracr an ideal i in a commutative noetherian ring r is a gorenstein ideal of grade g if pdr r i grade i g and the canonical module extr r i r is cyclic serre showed that if g 2 then i is a complete intersection and buchsbaum and eisenbud proved a structure theorem for the case g 3 we present generic resolutions for a class of gorenstein ideals of grade 4 and we illustrate the structure of the resolution with various specializations among these examples there are gorenstein ideals of grade 4 in k x y z v that are n generated for any odd integer n 7 we construct other examples from almost complete intersections of grade 3 and their canonical modules in the generic case the ideals are shown to be normal primes finally we conclude by giving an explicit associative algebra structure for the resolutions it is this algebra structure that we use to classify the different gorenstein ideals of grade 4 and which may be the key to a complete structure theorem

deformation and linkage of gorenstein algebras general double linkage of gorenstein algebras is defined rigidity genericity and regularity up to codimension six all pass across general double linkage rigid strongly unobstructed codimension four gorenstein algebras which lie in different herzog classes are produced introduction we begin by reviewing the classical structure theorems let p k xi x 41 a p i and f be the minimal resolution of a by free p modules f 1 if a has codimension 1 and i is unmixed then f is given by 0 p p for some fin p 2 hilbert burch 7 schaps 28 if a has codimension 2 and is cohen macaulay then f is given by x x 0 p n i ppnp where xis an n x n 1 matrix xi l 1l i x and a i x is the determinant formed by omitting the ith row from x 3 buchsbaum and eisenbud 4 if a has codimension 3 and its gorenstein then f is given by xt x x o p pn pn p where xis an n x n alternating matrix xi l i lpfi x and pfi x denotes the pfaffian of the alternating submatrix of x formed by omitting row and column i in these structure theorems as the codimension of a increases so does the number of conditions on a it is reasonable to suspect that a structure theorem for codimension four gorenstein algebras a will require that an additional hypothesis be imposed on a we observe that in each case a there is a structure theorem b all deformations are unobstructed see herzog 13 and c each algebra a is in the linkage class of a complete intersection see peskine and szpiro 27 and watanabe 30 historically these three facts were found independently with hindsight we now see that b follows from c and a follows in turn from b in his thesis received by the editors june 10 1982 and in revised form march 29 1983 some of the contents of this paper were presented to the special session on commutative algebra at the american mathematical society meeting in cincinnati on january 13 1982 1980 mathematics subject classification primary 13c05 13d25 13h10 secondary 14b07 14d115 14m05 16ao3 both authors were supported in part by the national science foundation 1984 american mathematical society 0002 9947 84 1 00 25 per page

infinite conservation laws in the one dimensional hubbard model we show that the hamiltonian of the 1d hubbard model commutes with a one parameter family of transfer matrices of a new 2d classical model corresponding to two coupled six vertex models central to this result is a new local algebraic relation a generalization of the infinitesimal startriangle relation

rotationally invariant ensembles of integrable matrices we construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory imt a counterpart of random matrix theory rmt for quantum integrable models a type m family of integrable matrices consists of exactly n m independent commuting n n matrices linear in a real parameter we first develop a rotationally invariant parametrization of such matrices previously only constructed in a preferred basis for example an arbitrary choice of a vector and two commuting hermitian matrices defines a type 1 family and vice versa higher types similarly involve a random vector and two matrices the basis independent formulation allows us to derive the joint probability density for integrable matrices similar to the construction of gaussian ensembles in the rmt

finite size scaling for correlations of quantum spin chains at criticality we study the finite size scaling behavior of two point correlation functions of translationally invariant many body systems at criticality we propose an efficient method for calculating the two point correlation functions in the thermodynamic limit from numerical data of finite systems our method is most effective when applied to a two dimensional classical system which possesses a conformal invariance by using this method with numerical data obtained from exact diagonalizations and monte carlo simulations we study the spin spin correlations of the quantum spin 1 2 and 3 2 antifierromagnetic chains in particular the logarithmic corrections to power law decay of the correlation of the spin 1 2 isotropic heisenberg antiferromagnetic chain are studied thoroughly we clarify the cause of the discrepancy in previous calculations for the logarithmic corrections our result strongly supports the field theoretic prediction based on the mappings to the wess zumino witten nonlinear model or the sine gordon model we also treat logarithmic corrections and crossover phenomena in the spin spin correlation of the spin 3 2 isotropic heisenberg antiferromagnetic chain our results are consistent with the affleck haldane prediction that the correlation of the spin 3 2 chain exhibits a crossover to the same asymptotic behavior as in the spin 1 2 chain

coloured frt algebra and its yang baxterization leading to integrable models with non additive r matrices a colour representation of faddeev reshetikhin takhtajan frt algebra which in contrast to the standard case is related to the coloured braid group representation with generic values of q is presented explicit realizations of l or matrices occurring in this coloured variant of frt algebra are also obtained for the uq gl 2 quantized algebra though these realizations are found to depend manifestly on the colour parameters the underlying quantum group structure and associated co product are interestingly free from such dependence this allows us to perform the yang baxterization of the coloured frt algebra successfully which leads to the construction of an ancestor law operator associated with a new non additive type quantum r matrix through different realizations of this lax operator a new class of quantum integrable models representing colour generalizations of the well known models such as the lattice sine gordon model the ablowitz ladik model lattice and the derivative nonlinear schrodinger model etc is generated

on the cubic lattice green functions wheatstone physics laboratory king s college university of london strand london wc2r 2ls u k it is proved that k k 4 eta frac 1 2 1 eta frac 1 2 k k where eta is a complex variable which lies in a certain region scr r 2 of the eta plane and k k pm are complete elliptic integrals of the first kind with moduli k pm which are given by k pm 2 equiv k pm 2 eta frac 1 2 pm frac 1 4 eta 4 eta frac 1 2 frac 1 4 2 eta 1 eta frac 1 2 this basic result is then used to express the face centred cubic and simple cubic lattice green functions at the origin in terms of the square of a complete elliptic integral of the first kind several new identities involving the heun function f a b alpha beta gamma delta eta are also derived next it is shown that the three cubic lattice green functions all have parametric representations which involve the green function for the two dimensional honeycomb lattice finally the results are applied to a variety of problems in lattice statistics in particular a new simplified formula for the generating function of staircase polygons on a four dimensional hypercubic lattice is derived

exact solution of an n sublattice vertex model the exact diagonalization of the diagonal to diagonal transfer matrix of an n sublattice vertex model with interactions of the vertex arrow type between sublattices is carried out using the generalized nested bethe ansatz method an exact expression for the free energy is obtained involving the solution of an integral equation which is studied numerically

on the cubic modular transformation and the cubic lattice green functions the elliptic modular transformation of order 3 is used to prove that the lattice green functions at the origin for the face centred cubic and simple cubic lattices can both be expressed in terms of the square of a complete elliptic integral of the first kind

algebraic bethe ansatz for a class of coupled asymmetric six vertex free fermion model we present an algebraic bethe ansatz for certain submanifolds of the bilayer vertex models proposed by shiroishi and wadati as coupled asymmetric six vertex free fermion models a peculiar feature of our formulation is the presence of a diagonal monodromy matrix element that does not generate unwanted terms the model contains two free parameters entering into the bethe ansatz equations as a pure phase factor

the graded yang baxter relation for the one dimensional bariev chain of fermions the graded version of quantum yang baxter relation for the one dimensional bariev chain of fermions is formulated from this relation the commutativity of the transfer matrices for different values of the spectral parameter follows our result is consistent with the applicability of the traditional bethe ansatz method further the first non trivial conserved current next to the hamiltonian is constructed

derivation of exact product forms for the simple cubic lattice green function using fourier generating functions and lie group identities the simple cubic lattice green function is investigated where l1 l2 l3 is a set of integers w w1 iw2 is a complex variable and is a real parameter in the interval 0 in particular a new and direct method is used to prove that g 2n n n w and g n n n 1 w can be expressed in terms of a product of two 2f1 hypergeometric functions where n 0 1 2 in earlier work delves and joyce obtained these 2f1 product forms by solving complicated fourth order linear differential equations of the fuchsian type in this paper fourier generating functions and a known lie group addition theorem play crucial roles in the derivation of the product forms many term recurrence relations are also derived for g 2n n n w and g n n n w

integrability of the one dimensional bariev model we investigate the exact integrability of the one dimensional 1d bariev model in the framework of the quantum inverse scattering method qism using the jordan wigner transformation the 1d bariev model can be regarded as a coupled spin model we construct the higher conserved currents which commute with the hamiltonian the explicit form of the conserved currents helps us to infer the l operator of the 1d bariev model from the l operator we construct a transfer matrix which is a generating function of the conserved currents we also find the corresponding r matrix which satisfies the yang baxter relation thus the exact integrability of the 1d bariev model is established the r matrix does not have the difference property for the spectral parameter as in the case of the 1d hubbard model we also provide the lax representation and the fermionic formulation of the yang baxter relation

exact values for the cubic lattice green functions it is shown that the value of the cubic lattice green functions g z l m n at the upper critical point z 3 for the simple cubic and face centred cubic lattices or z 1 for the body centred cubic lattice is expressible rationally in terms of the known value of g 3 0 0 0 or g 1 0 0 0 and

non additive fusion hubbard models and non locality within the framework of quantum groups and additive r matrices the fusion procedure allows one to construct higher dimensional solutions of the yang baxter equation these solutions lead to integrable one dimensional spin chain hamiltonians here fusion is shown to generalize naturally to non additive r matrices which therefore do not have a quantum group symmetry this method is then applied to the generalized hubbard models although the resulting integrable models are not as simple as the starting ones the general structure is that of two spin s s sl 2 models coupled at the free fermion point an important issue is the probable lack of regular points which give local hamiltonians this problem is related to the existence of second order zeros in the unitarity equation and arises for the xx models of higher spins the building blocks of the hubbard models a possible connection between some lax operators l and r matrices is noted

the minimal resolution of a codimension four almost complete intersection is a dg algebra abstract let r m k be a commutative noetherian local ring in which two is a unit we prove that if j is a five generated grade four perfect ideal in r then the minimal resolution of r j by free r modules is an associative differential graded commutative algebra this result extends and completes the work in j algebra 159 1993 1 46 where the conclusion is shown to hold provided certain technical conditions on tor are satisfied the multiplication on the resolution of r j is constructed using appropriate higher order multiplication on the resolution of r i where i is a gorenstein ideal which is linked to j

integrable variant of the one dimensional hubbard model a new integrable model which is a variant of the one dimensional hubbard model is proposed the integrability of the model is verified by presenting the associated quantum r matrix which satisfies the yang baxter equation we argue that the new model possesses the so 4 algebra symmetry which contains a representation of the h pairing su 2 algebra and a spin su 2 algebra additionally the algebraic bethe ansatz is studied by means of the quantum inverse scattering method the spectrum of the hamiltonian eigenvectors as well as the bethe ansatz equations are discussed 2002 american institute of physics doi 10 1063 1 1481956

the supersymmetric t j model in one dimension the t j model related to the strong correlation limit of the hubbard model is shown to be soluble in one dimension using the bethe ansatz the solution holds only when the hamiltonian is supersymmetric the ground state in the presence of holes is found to be gapless and to have no magnetization

new exact integrable spin 1 quantum chains new exactly integrable quantum spin 1 chains are derived these new quantum hamiltonians like the xxz chain have a free parameter anisotropy and commute with the z component of the total magnetization the eigenvalues and eigenvectors are obtained directly by imposing a generalized coordinate bethe ansatz the derived bethe ansatz equations have an unusual form and the associated r matrix has a dependence with the spectral parameter that is not of difference form like the hubbard quantum chain

a class of parameter dependent commuting matrices we present a novel class of real symmetric matrices in arbitrary dimension d linearly dependent on a parameter x the matrix elements satisfy a set of nontrivial constraints that arise from asking for commutation of pairs of such matrices for all x and an intuitive sufficiency condition for the solvability of certain linear equations that arise therefrom this class of matrices generically violates the wigner von neumann non crossing rule and is argued to be intimately connected with finite dimensional hamiltonians of quantum integrable systems

singular behaviour of the cubic lattice green functions and associated integrals the behaviour of the three cubic lattice green functions where is determined in the neighbourhood of the singular point w 1 the results are used to investigate the singular behaviour of the associated integrals finally a new method is developed which enables one to calculate the numerical values of the constants lj 1 j 1 2 3 with extremely high precision

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

exact evaluation of the resistance in an infinite face centered cubic network the equivalent resistance between the origin and the lattice site 2n 0 0 in an infinite face centered cubic fcc network consisting from identical resistors each of resistance r has been evaluated analytically and numerically the asymptotic behavior of the equivalent resistance has been also investigated finally some numerical values for the equivalent resistance are presented

decorated star triangle relations and exact integrability of the one dimensional hubbard model the exact integrability of the one dimensional hubbard model is demonstrated with the help of a novel set of triangle relations the decorated star triangle relations the covering two dimensional statistical mechanical model obeys the star triangle or yang baxter relation a conjecture is presented for the eigenvalues of the transfer matrix

the half filled hubbard model in the pair approximation of the cluster variation method the half filled hubbard model is studied in the pair approximation of the cluster variation method cvm the use of the so 4 symmetry of the model makes it possible to give a complete analytical characterization of the ground state by means of explicit expressions for the double occupancy and the nearest neighbour correlation functions the finite temperature analysis is reduced to the numerical solution of only two coupled transcendental equations the behaviour of local magnetic moment specific heat and correlation functions is given for some typical cases in one and two dimensions the authors obtain good qualitative agreement with exact and numerical results in one dimension the results for finite temperatures show a rapid evolution with increasing temperature from a strongly antiferromagnetic behavior to a disordered one in the high temperature region a maximum which has been related to a gradual metal insulator transition is found in the specific heat for very large values of the coulomb repulsion

yang baxter equation for the r matrix of the 1d su n hubbard model based on the tetrahedral zamolodchikov algebra we prove the yang baxter equation for the r matrix of the 1d su n hubbard model furthermore we present generalizations of the model

exact spectrum analysis of a one dimensional hubbard model for a one dimensional hubbard model with a half filled band and one extra hole the single hole green function is calculated analytically the spectrum analysis shows that the n electrons tend to move so as to form identical and consecutive packets to satisfy rules fixed by arithmetical properties of the integer n

large parameter cases of the gauss hypergeometric function we consider the asymptotic behavior of the gauss hypergeometric function when several of the parameters a b c are large we indicate which cases are of interest for orthogonal polynomials jacobi but also meixner krawtchouk etc which results are already available and which cases need more attention we also consider a few examples of 3f2 functions of unit argument to explain which difficulties arise in these cases when standard integrals or differential equations are not available

gorenstein rings as specializations of unique factorization domains it is known that a unique factorization domain which is a cohenmacaulay factor ring of a regular local ring is gorenstein 221 see however 23 in this paper we show that conversely any gorenstein ring r which is a factor ring of a regular local ring and a complete intersection locally in codimension one can be realized as a specialization of a unique factorization domain s which is cohen macaulay and an epimorphic image of a regular local ring proposition 1 s is obtained as a local blowing up ring of a generic link of r for the definitions we refer to the first section of this paper as a first application we prove that for certain singularities the highest exterior power and any symmetric power of the module of differentials are never cohen macaulay corollary 1 in a second corollary we construct examples of bad unique factorization domains s such that each s is a local cohen macaulay ring and the singular locus of s has big codimension but nevertheless the completion of s is no longer factorial this contrasts with a result of h flenner 7 1 51 saying essentially that for quasihomogeneous singularities which fulfill serre s conditions s and r2 factoriality is preserved under completion now let r be the reduced local ring of a quasihomogeneous gorenstein singularity over a perfect field such that r is a complete intersection locally in codimension two and the degrees of the homogeneous free resolution of r are big enough by s we denote the singularity constructed in proposition 1 in this situation we can use flenner s result to conclude that also the completion s of s is factorial and hence z admits a formal deformation to the complete unique factorization domain s this implies that rigid gorenstein singularities satisfying the above assumptions are factorial proposition 2 129 0021 8693 84 3 00

licci squarefree monomial ideals generated in degree two or with deviation two abstract we study squarefree monomial ideals which are in the linkage class of a complete intersection we call such an ideal licci firstly we classify all licci edge ideals secondly we prove that any cohen macaulay almost complete intersection squarefree monomial ideal is licci thirdly we characterize licci squarefree monomial ideals of deviation 2 in terms of hypergraphs associated to the ideals as an application of the classification in kimura terai and yoshida 2009 15

linkage theory for algebras with pure resolutions abstract let s be a graded cohen macaulay quotient r i of a polynomial ring r k x 1 x n over an infinite field k if s is in the linkage class of a complete intersection then its koszul homology and twisted conormal module are cohen macaulay by passing to zero and one dimensional specializations of s one can convert these properties into numerical test conditions for a graded algebra to be in the linkage class of a complete intersection we prove that if s is generically a complete intersection and the minimal free solution of s is pure and almost linear then s is not in the linkage class of a complete intersection apart from a few obvious exceptions for algebras of small codimension the results are stronger if s has codimension three and has a pure resolution then s is in the linkage class of a complete intersection if and only if s is gorenstein if s is a codimension four gorenstein algebra with a pure resolution then s is in the linkage class of a complete intersection if and only if s is a complete intersection

symmetries and reduced systems of equations for three boson and four boson interactions a reduced system of equations is found for three boson and four boson interactions the invariants and surfaces in the space of parameters connected with these systems are constructed it is shown that at these surfaces the eigenvectors can be presented in power form the connection between the ansatz and polynomial extensions of lie algebras is briefly discussed the computational procedures corresponding to the original equations and the method of polynomial algebras are compared

coordinate bethe ansatz for the one dimensional su n hubbard model with open boundary conditions the one dimensional 1d su n hubbard model with open boundary condition is solved by using the coordinate bethe ansatz method the energy and integrable boundary conditions are obtained at the same time the corresponding bethe ansatz equations are achieved by diagonalizing the inhomogeneous transfer matrix of the open su 2n 2 xxx vertex model when n 2 our result comes back to that of the 1d hubbard model given by deguchi and yue con mat 9704138

exact solution of generalized t j models in one dimension a relation between the anisotropic t j model and a generalized six vertex model has been established on the basis of this relation we obtain the exact bethe ansatz solution for the ground state energy of anisotropic t j model for arbitrary spin and special values of coupling parameters we consider also the anisotropic generalization of t j model with localons

poincare series of modules over local rings of small embedding codepth or small linking number abstract let s n k be a commutative noetherian local ring and m be a finitely generated s module then the poincare series of m is rational if any of the following holds edim s depth s 3 edim s depth s 4 and s is gorenstein s is one link from a complete intersection s is two links from a complete intersection and s is gorenstein to prove this result we assume without loss of generality that s r i with r m regular local and the key to the argument is to produce a factorization of the canonical map r s as a composition of a complete intersection r c with a golod map c s this is accomplished by invoking a theorem of avramov and backelin once one has existence of a dg algebra structure on a minimal r free resolution of s together with detailed knowledge of the structure of the induced homology algebra tor r s k h k s linkage theory provides the main technical tool for this analysis

integrability test for spin chains we examine a simple heuristic test of integrability for quantum chains this test is applied to a variety of systems including isotropic spin 1 models with nearest neighbour interaction potts type inhomogeneous spin 1 2 chains and a multiparameter family of spin 1 2 models generalizing the xyz chain with next to nearest neighbour interactions and bond alternation within the latter family we determine all the integrable models with an o 2 symmetry

exact integrability of the one dimensional hubbard model the 1d hubbard model is shown to be an exactly integrable system a covering model of 2d statistical mechanics which i proposed recently was shown to provide a one parameter family of transfer matrices commuting with the hamiltonian of the hubbard model i show in this work that any two transfer matrices of a family commute mutually at the root of the commutation relation is the ubiquitous yang baxter factorization condition the form of the r operator is displayed explicitly

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

lax pair formulation for one dimensional hubbard open chain with chemical potential the lax pair for the one dimensional hubbard spin chain with open boundary conditions bc is explicitly constructed and two different kinds of boundary k matrices compatible with the integrability of the model are obtained our construction provides an alternative and direct demonstration for the quantum integrability of the model and it is found that the model is related to a class of commuting transfer matrices of an equivalent coupled asymmetric six vertex model with open bc our results show that the chemical potential term is indeed nontrivial for the underlying algebraic structure of the model

numerical invariants of liaison classes 302 c huneke intersection what properties are then enjoyed by v for instance buchweitz was able to show that all such v are strongly nonobstructed see 5 for details the two questions above have been completely answered if x spec r with r a regular local ring and if codim v 2 in this case

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give

hidden local gauge invariance in the one dimensional hubbard model and its equivalent coupled spin model hidden local gauge invariance in the one dimensional 1 d hubbard model and its equivalent coupled spin model is studied it is found that abelian u 1 u 1 gauge transformations appear in both cases furthermore it is shown that the energy spectrum is gauge invariant whereas the eigenvectors are explicitly gauge dependent however this result relies heavily on shastry s conjecture about the eigenvalue of the transfer matrix for the 1 d hubbard model lastly there is also a discrete symmetry associated to z2 z2 once this symmetry is broken one immediately obtains another nontrivial solution to the yang baxter relations ufaipxr

the minimal free resolutions of the huneke ulrich deviation two gorenstein ideals structure theorems exist for gorenstein ideals of small grade a different parameter one can use for organizing ideals is deviation or complete intersection defect the deviation of the ideal i is the minimal number of generators of i minus the grade of i a deviation zero ideal is a complete intersection kunz s proved that there are no deviation one gorenstein ideals presently people are trying to find all deviation two gorenstein ideals see for example 3 a hypersurface section of a grade g 1 deviation two gorenstein ideal is a grade g deviation two gorenstein ideal we consider this to be a trivial example very few non trivial examples of deviation two gorenstein ideals are known the buchsbaum eisenbud structure theorem 2 describes all grade three gorenstein ideals in particular every deviation two grade three gorenstein ideal is generated by the maximal order pfaflians of a 5 x 5 alternating matrix no non trivial deviation two gorenstein ideals of even grade are known herzog and miller 3 have taken a step toward proving that all grade four deviation two gorenstein ideals are trivial let i be a grade four deviation two gorenstein ideal if i is a generic complete intersection and i i is cohenmacaulay then i is a hypersurface section the only known non trivial deviation 2 gorenstein ideals of grade at least four are the ideals of huneke and ulrich 4 which are the ideals studied in this paper for each integer s let y be a 1 x s matrix of indeterminates x be an s x s alternating matrix of indeterminates g be the product y x and ct be the sequence gp g let a be the ideal of

exact evaluation of the green functions for the anisotropic face centred and simple cubic lattices the analytic properties of the lattice green functions and are investigated where w1 w2 are complex variables and 1 2 are real parameters in the interval 0 in particular simple and direct methods are developed which enable one to evaluate g1 1 w1 and g2 2 w2 in terms of products of two complete elliptic integrals of the first kind kampe de feriet series are also used to derive new transformation formulae which give connections between g1 1 w1 and g2 2 w2

sums of linked ideals it is shown that the sum of two geometrically linked ideals in the linkage class of a complete intersection is again an ideal in the linkage class of a complete intersection conversely every gorenstein ideal of height at least two in the linkage class of a complete intersection can be obtained as a generalized localization of a sum of two geometrically linked ideals in the linkage class of a complete intersection we also investigate sums of doubly linked gorenstein ideals as an application we construct a perfect prime ideal which is strongly nonobstructed but not strongly cohen macaulay and a perfect prime ideal which is not strongly nonobstructed but whose entire linkage class is strongly cohen macaulay

classification of parameter dependent quantum integrable models their parameterization exact solution and other properties we study general quantum integrable hamiltonians linear in a coupling constant and represented by finite n n real symmetric matrices the restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into types according to the number of such integrals a type m family in our definition is formed by n m nontrivial mutually commuting operators linear in the coupling working from this definition alone we parameterize type m operators i e resolve the commutation relations and obtain an exact solution for their eigenvalues and eigenvectors we show that our parameterization covers all type 1 2 and 3 integrable models and discuss the extent to which it is complete for other types we also present robust numerical observation on the number of energy level crossings in type m integrable systems and analyze the taxonomy of types in the 1d hubbard model

on the exact evaluation of the face centred cubic lattice green function the mathematical properties of the lattice green function open image in new window are investigated where w w 1 iw 2 lies in a complex plane which is cut from w 1 to w 3 and l 1 l 2 l 3 is a set of integers with l 1 l 2 l 3 equal to an even integer in particular it is proved that g 2n 0 0 w where n 0 1 2 is a solution of a fourth order linear differential equation of the fuchsian type with four regular singular points at w 1 0 3 and it is also shown that g 2n 0 0 w satisfies a five term recurrence relation with respect to the integer variable n the limiting function g 2n 0 0 w 1 equiv lim epsilon rightarrow0 g 2n 0 0 w 1 mathrm i epsilon g mathrm r 2n 0 0 w 1 mathrm i g mathrm i 2n 0 0 w 1 nonumber where w 1 1 3 is evaluated exactly in terms of 2 f 1 hypergeometric functions and the special cases g 2n 0 0 0 g 2n 0 0 1 and g 2n 0 0 3 are analysed using singular value theory more generally it is demonstrated that g l 1 l 2 l 3 w can be written in the form open image in new window where open image in new window are rational functions of the variable k k and e k are complete elliptic integrals of the first and second kind respectively with k 2 equiv k 2 w 1 over2 2 over w biggl 1 1 over w biggr 3 over2 1 over2 biggl 1 1 over w biggr biggl 1 1 over w biggr 3 over2 biggl 1 3 over w biggr 1 over2 nonumber and the parameter is defined as xi equiv xi w biggl 1 sqrt 1 3 over w biggr 1 biggl 1 sqrt 1 1 over w biggr nonumber this result is valid for all values of w which lie in the cut plane the asymptotic behaviour of g 2n 0 0 w 1 and g 2n 0 0 w 1 as n is also determined in the final section of the paper a new 2 f 1 product form for the anisotropic face centred cubic lattice green function is given