theta function parametrization and fusion for 3d integrable boltzmann weights we report progress in constructing boltzmann weights for integrable threedimensional lattice spin models we show that a large class of vertex solutions to the modified tetrahedron equation mte can be conveniently parametrized in terms of nth roots of theta functions on the jacobian of a compact algebraic curve fay s identity guarantees the fermat relations and the classical equations of motion for the parameters determining the boltzmann weights our parametrization allows us to write a simple formula for fused boltzmann weights r which describe the partition function of an arbitrary open box and which also obey the modified tetrahedron equation imposing periodic boundary conditions we observe that the r satisfy the normal tetrahedron equation the scheme described contains the zamolodchikov baxter bazhanov model and the chessboard model as special cases

on global small amplitude solutions to systems of cubic nonlinear klein gordon equations with different mass terms in one space dimension abstract we consider the cauchy problem for systems of cubic nonlinear klein gordon equations with different mass terms in one space dimension we prove some result concerning the global existence of small amplitude solutions and their asymptotic behavior as a consequence we see that the condition for small data global existence is actually influenced by the difference of masses in some cases

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

an integrable model with a non reducible three particle r matrix we define an integrable lattice model which in the notation of yang in addition to the conventional two particle r matrices also contains non reducible three particle r matrices the corresponding modified yang baxter equations are solved and an expression for the transfer matrix is found as a normal ordered exponential of a non local hamiltonian

symmetry relations in exactly soluble models symmetry relations such as the star triangle or the inverse relation are very useful in determining the partition function of two dimensional exactly soluble models a common construction of the three dimensional equivalents of these symmetry relations is presented they are used to derive in a geometric way applying simultaneously to different kinds of spin models the consequent global properties i e the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function the usefulness of the inverse relation is illustrated by an application to the three dimensional ising model

three dimensional integrable models and associated tangle invariants in this paper we show that the boltzmann weights of the three dimensional baxter bazhanov model give representations of the braid group if some suitable spectral limits are taken in the trigonometric case we classify all possible spectral limits which produce braid group representations furthermore we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones

on algorithmic coxeter spectral analysis of positive posets abstract following a general framework of coxeter spectral analysis of signed graphs and finite posets i introduced by simson siam j discrete math 27 827 854 2013 we present efficient numerical algorithms for the coxeter spectral study of finite posets i 1 n i that are positive in the sense that the symmetric gram matrix g i 1 2 c i c i t r m n q is positive definite where c i m n z is the incidence matrix of i encoding the relation i in the framework of scientific computing we present a complete coxeter spectral classification of finite positive posets i of size n i 20 it extends one of the main results obtained in gasiorek et al eur j comb 48 127 142 2015 for posets of size n 10 we also show that the connectivity of such posets i is determined by the complex coxeter spectrum specc i c equivalently by the coxeter polynomial cox i t z t od i one of the main results of the paper is a new technique to compute in a polynomial time a z invertible matrix b i m n z such that b i t r c i b i g d i where g d i m n z is a non symmetric gram matrix of a simply laced dynkin diagram d i a n d n e 6 e 7 e 8 associated with a finite positive poset i

simplex equations and their solutions we investigate ann simplex generalization of the classical and quantum yang baxter equation for the case ofsl 2 we find the most general solution of the classicaln simplex equation for alln these classical solutions can be quantized in the sense of quantum group theory forn 2 3 and we exhibit a quantum solution to the tetrahedron equations n 3 the classical nondegenerate solutions cannot be quantized forn 4

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

hyperbolic coxeter groups for triangular potts models the symmetry groups generated by the inversion relations of lattice models of statistical mechanics on triangular lattices are analysed for vertex models and for the standard scalar potts model with two and three site interactions these groups are generated by three inversion relations and are seen to be generically very large ones hyperbolic groups two situations for which the representations of these groups degenerate into smaller ones hopefully compatible with integrability are considered the first reduction for the vertex triangular model corresponds to the situation where the vertex of the triangular model coincides with the left or right hand side of a yang baxter relation in this case the representation of the group is isomorphic up to a semi direct product by a finite group to z z the second reduction for q state ports models occurs for particular values of q the so called tutte beraha numbers for this model algebraic varieties including the known ferromagnetic critical variety happen to be invariant under such large groups of symmetries as a byproduct this analysis provides nice birational representations of hyperbolic coxeter groups

quantization of three wave equations the subject of this paper is the revision of discretization and quantization of two similar classical integrable hamiltonian systems in three dimensional spacetime the standard three wave equations and the modified three wave equations the darboux manakov zakharov system in two distinguished gauges the quantized systems in discrete spacetime may be understood as the regularized integrable quantum field theories integrability of the theories and in particular the quantum tetrahedron equations for vertex operators follow from the quantum auxiliary linear problems the principal object of the lattice field theories is the heisenberg discrete time evolution operator constructed with the help of vertex operators

trigonometry of spacetimes a new self dual approach to a curvature signature in dependent trigonometry a new method to obtain trigonometry for the real spaces of constant curvature and metric of any even degenerate signature is presented the method could be described as curvature signature in dependent trigonometry and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation this brings to its logical end the idea of an absolute trigonometry and provides equations which hold true for the nine two dimensional spaces of constant curvature and any signature this family of spaces includes both relativistic and non relativistic spacetimes therefore a complete discussion of trigonometry in the six de sitter minkowskian newton hooke and galilean spacetimes follow as particular instances of the general approach distinctive traits of the method are universality and self duality every equation is meaningful for the nine spaces at once and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves these basic structural properties allow a complete study of trigonometry and in fact any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes in most cases these equations are new

the eight tetrahedron equations in this paper we derive from arguments of string scattering a set of eight tetrahedron equations with different index orderings it is argued that this system of equations is the proper system that represents integrable structures in three dimensions generalizing the yang baxter equation under additional restrictions this system reduces to the usual tetrahedron equation in the vertex form most known solutions fall under this class but it is by no means necessary comparison is made with the work on braided monoidal 2 categories also leading to eight tetrahedron equations

functional relations and nested bethe ansatz 3 chiral potts model at q 2 1 we obtain the functional relations for the eigenvalues of the transfer matrix of the sl 3 chiral potts model for for the homogeneous model in both directions a solution of these functional relations can be written in terms of roots of bethe ansatz like equations in addition a direct nested bethe ansatz has also been developed for this case

dilogarithm identities in conformal field theory and group homology recently rogers dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations one of the connecting threads is an identity of richmond szekeres that appeared in the computation of central charges in conformal field theory we show that the richmond szekeres identity and its extension by kirillov reshetikhin equivalent to an identity found earlier by lewin can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all 2 2 real matrices viewed as adiscrete group this connection allows us to clarify a few of the assertions and conjectures stated in the work of nahm recknagel terhoven concerning the role of algebraic k theory and thurston s program on hyperbolic 3 manifolds specifically it is not related to hyperbolic 3 manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all 2 2 real matrices viewed as a topological group this also resolves the weaker version of the conjecture as formulated by kirillov we end with a summary of a number of open conjectures on the mathematical side

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

localising embeddings of comodule categories with applications to tame and euler coalgebras given a k coalgebra c and an injective left c comodule e we construct a coalgebra ce and fully faithful left exact embedding e ce comod c comod of comodule categories such that the image of e is the subcategory c comode consisting of the comodules m with an injective presentation 0 m e0 e1 where e0 and e1 are direct sums of direct summands of the comodule e the functor e preserves the indecomposability the injectivity and is right adjoint to the restriction functor rese c comod ce comod applications to the study of tame coalgebras betti numbers and cosyzygy comodules of simple comodules over a left euler coalgebra c are given a localising reduction to countably dimensional euler coalgebras is presented

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

abelian categories almost split sequences and comodules the following are equivalent for a skeletally small abelian horn finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length a each indecomposable injective has a simple subobject b the category is equivalent to the category of socle finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle finitely copresented c the category has left almost split sequences

two state spectral free solutions of the frenkel moore simplex equation whilst many solutions have been found for the quantum yang baxter equation qybe there are fewer known solutions available for its higher dimensional generalizations zamolodchikov s tetrahedron equation zte and frenkel and moore s simplex equation fme in this paper we present families of solutions to the fme which may help us to understand more about higher dimensional generalization of the qybe

on pentagon ten term and tetrahedron relations it is shown that the tetrahedron equation under the substitution where p 23 is the permutation operator is reduced to a pair of pentagon and one ten term equations on operators s and examples of infinite dimensional solutions are found o doubles of novikov which generalize the heisenberg double of a hopf algebra provide a particular algebraic solution to the problem

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 vertex formulation of the bazhanov baxter model in this paper we formulate an integrable model on the simple cubic lattice then valued spin variables of the model belong to edges of the lattice the boltzmann weights of the model obey the vertex type tetrahedron equation in the thermodynamic limit our model is equivalent to the bazhanov baxter model in the case whenn 2 we reproduce korepanov s and hietarinta s solutions of the tetrahedron equation as special cases

star triangle relations in the exactly solvable statistical models a regular method for analysis of lattice spin models with a nearest neighbour interaction is proposed star triangle relations in the form of functional equations are used parametric families of transfer matrices commuting due to star triangle relations are constructed the eigenvalues of transfer matrices as functions of the spectral parameter are shown to obey two functional equations the solution of these equations for the maximal eigenvalue yields the partition function of the model the method is applied for evaluation of the partition function of the critical potts models the ising model the ashkin teller model equivalent to the eight vertex model

towards an exhaustive classification of the star triangle relation ii for pt i see ibid vol 17 p 1251 1984 the authors continue their study of the star triangle relation for two component spin models with the help of the inversion relation the conclusion is that the most general interaction round a face models 16 non zero parameters that satisfy the star triangle relations are mainly baxter 1972 and fan and wu 1970 models in this way they recover krichever s result 1981 for the classification of the star triangle relations for vertex models they sketch a generalisation of this approach to models with excluded configurations to staggered models and to three dimensional models tetrahedron relations

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

remarks on the star triangle relation in the baxter bazhanov model we show that the restricted star triangle relation introduced by bazhanov and baxter can be obtained either from the star triangle relation of the chiral potts model or from the star square relation proposed by kashaev mangazeev and stroganov and give a response to a guess of bazhanov and baxter

an integrable 3d lattice model with positive boltzmann weights in this paper we construct a three dimensional 3d solvable lattice model with non negative boltzmann weights the spin variables in the model are assigned to edges of the 3d cubic lattice and run over an infinite number of discrete states the boltzmann weights satisfy the tetrahedron equation which is a 3d generalization of the yang baxter equation the weights depend on a free parameter 0 q 1 and three continuous field variables the layer to layer transfer matrices of the model form a two parameter commutative family this is the first example of a non trivial solvable 3d lattice model with non negative boltzmann weights

permutation type solutions to the yang baxter and other n simplex equations we study permutation type solutions to n simplex equations that is solutions whose matrix form can be written as with some matrix a and vector b both over with this ansatz the equations of the n simplex equation reduce to a matrix equation over we have completely analysed the 2 3 and 4 simplex equations in the generic d case the solutions show interesting patterns that seem to continue to still higher simplex equations

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

quantum 2 1 evolution model a quantum evolution model in 2 1 discrete spacetime connected with a 3d fundamental map is investigated map is derived as a map providing a zero curvature of a 2d linear lattice system called the current system in a special case of the local weyl algebra for dynamical variables the map appears to be canonical and it corresponds to the known operator valued matrix the current system is a type of the linear problem for the 2 1 evolution model a generating function for the integrals of motion for the evolution is derived with the help of the current system thus the complete integrability in 3d is proved directly

statistical mechanics a perspective on the work of r j baxter abstract the work of rodney baxter in statistical mechanics is surveyed with an emphasis on the role that his exact calculations have played in the development of the modern theory of critical phenomena

star triangle relation for a three dimensional model the solvablesl n chiral potts model can be interpreted as a three dimensional lattice model with local interactions to within a minor modification of the boundary conditions it is an ising type model on the body centered cubic lattice with two and three spin interactions the corresponding local boltzmann weights obey a number of simple relations including a restricted star triangle relation which is a modified version of the well known star triangle relation appearing in two dimensional models we show that these relations lead to remarkable symmetry properties of the boltzmann weight function of an elementary cube of the lattice related to the spatial symmetry group of the cubic lattice these symmetry properties allow one to prove the commutativity of the row to row transfer matrices bypassing the tetrahedron relation the partition function per site for the infinite lattice is calculated exactly

on the solution of some vertex models using factorizable s matrices the intimate connection between factorizables matrices and some vertex models in two dimensions to be reviewed here is exploited to show that the knowledge of thes matrix not only allows us to define a solvable vertex modela la zamolodchikov but often to write down the free energy by inspection the prototype for discussion is baxter s eight vertex model generated by zamolodchikov s z4 s matrix the method is then applied to a hitherto unsolved 19 vertex model based on the isospin 1s matrix of zamolidchikov and fateev and agreement is checked to fourth order in a perturbation series the possibility of molding other problems like theq state potts model into this framework is considered

inverses of cartan matrices of lie algebras and lie superalgebras abstract we derive explicit formulas for the inverses of the cartan matrices of the simple lie algebras and the basic classical lie superalgebras as well as for their infinite generalizations

symmetries of the octonionic root system of e8 octonionic root system of e8 is decomposed as the 9 sets of hurwitz integers each set satisfying the binary tetrahedral group structure and the 12 sets of quaternionic units each set obeying the dicylic group structure of order 12 this fact is used to prove that the automorphism group of the octonionic root system of e7 is the finite subgroup of g2 of order 12 096 an explicit 7 7 matrix realization of which is constructed possible use of the octonionic representation of the e6 root system as orbifolds and the relevance of the binary tetrahedral structures with the statistical mechanics models are suggested

long range scattering for nonlinear schrodinger equations in one space dimension we consider the scattering problem for the nonlinear schrodinger equation in 1 1 dimensions open image in new window where x r 0 r p 3 we show that modified wave operators for exist on a dense set of a neighborhood of zero in the lebesgue spacel2 r or in the sobolev spaceh1 r the modified wave operators are introduced in order to control the long range nonlinearity u 2u

on an algebraic approach to higher dimensional statistical mechanics we study representations of temperley lieb algebras associated with the transfer matrix formulation of statistical mechanics on arbitrary lattices we first discuss a new hyperfinite algebra the diagram algebra d underset raise0 3em hbox smash scriptscriptstyle n q which is a quotient of the temperley lieb algebra appropriate for potts models in the mean field case and in which the algebras appropriate for all transverse lattice shapesg appear as subalgebras we give the complete structure of this subalgebra in the case hat a n potts model on a cylinder the study of the full temperley lieb algebra of graphg reveals a vast number of infinite sets of inequivalent irreducible representations characterized by one or more complex parameters associated to topological effects such as links we give a complete classification in the hat a n case where the only such effects are loops and twists

multidimensional inhomogeneous mixed initial boundary value problem for the nonlinear schrodinger equation abstract we consider the inhomogeneous mixed initial boundary value problem for the nonlinear multidimensional schrodinger equation formulated on upper right quarter plane we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time also we are interested in the study of the influence of the mixta boundary data on the asymptotic behavior of solutions our approach to get well posedness of nonlinear problems is based on the studying a linear theory and then using the fixed point argument to get a linear theory for the multidimensional model we propose general method based on laplace approach and theory cauchy type integral equations to get smooth solutions in l we modify a method based on the factorization for the free schrodinger evolution group the advantage of our approach is that it can also be applied to non integrable equations and arbitrary boundary conditions this approach is new and it is not standard we believe that the results of this paper could be applicable to study a wide class of dissipative multidimensional nonlinear equations by the use of techniques of nonlinear analysis

coxeter spectral classification of almost tp critical one peak posets using symbolic and numeric computations following an idea of spectral graph theory we give several characterizations of finite one peak posets i with almost p critical quadratic tits form q i zi z by applying linear algebra methods combinatorial algorithms an old idea of s a ovsienko and recent results by v m bondarenko and m v stepochkina our study is mainly motivated by an important application of the quadratic tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of nazarova roiter matrix problems over a field k in particular our study is inspired by a well known result of ju a drozd explained in introduction one of our main results asserts that given a finite poset i 1 n t such that is its unique maximal element and q i is almost p critical we have i the form q i is non negative and the subgroup kerq i v zn 1 q i v 0 of zi zn 1 is generated by a vector h h1 hn h with h1 0 hn 0 ii 5 i 9 and the coxeter polynomial coxi t z t coincides with the coxeter polynomial f i t of a uniquely determined euclidean diagram i d 4 e 6 e 7 e 8 iii the z bilinear tits form b i zn 1 zn 1 z of i is z equivalent to the gram z bilinear form b i zn 1 zn 1 z of the euclidean diagram i iv there is a z invertible matrix c such that c itr ctr c i c and c2 e where c i mn 1 z is the tits matrix of i and v i is one of the posets l1 l132 listed in the paper in particular we get in an alternative way the list of p critical posets t in the form t i presented by v m bondarenko and m v stepochkina 6 and we determine their coxeter euclidean types there are 115 one peak posets i l1 l115 with q i p critical and 17 exceptional posets i l116 l132 with q i almost p critical but not p critical our lists are obtained by using computer algebra tools mainly symbolic computation in maple and numeric computation in c

algebraic invariants in soluble models exhibits on the known solutions of the triangle relation for the exactly solved models some simplifying methods of recovering their parametrisation in terms of algebraic varieties the relation to the automorphy group generated by the inverse and spatial symmetries of the model is also analysed

thermodynamic limit for a spin lattice an integrable spin lattice is a higher dimensional generalization of integrable spin chains in this paper we consider a special spin lattice related to quantum mechanical interpretation of the three dimensional lattice model in statistical mechanics zamolodchikov and baxter the integrability means the existence of a set of mutually commuting operators expressed in the terms of local spin variables the significant difference between spin chain and spin lattice is that the commuting set for the latter is produced by a transfer matrix with two equitable spectral parameters there is a specific bilinear functional equation for the eigenvalues of this transfer matrix the spin lattice is investigated in this paper in the limit when both sizes of the lattice tend to infinity the limiting form of bilinear equation is derived it allows to analyze the distributions of eigenvalues of the whole commuting set the ground state distribution is obtained explicitly a structure of excited states is discussed

a classification of four state spin edge potts models we classify four state spin models with interactions along the edges according to their behaviour under a specific group of symmetry transformations this analysis uses the measure of complexity of the action of the symmetries in the spirit of the study of discrete dynamical systems on the space of parameters of the models and aims at uncovering solvable ones we find that the action of these symmetries has low complexity polynomial growth zero entropy we obtain natural parametrizations of various models among which an unexpected elliptic parametrization of the four state chiral potts model which we use to localize possible integrability conditions associated with high genus curves

hamiltonian limit of the 3d zamolodchikov model a two dimensional quantum hamiltonian n m commuting with the layer to layer transfer matrix of the three dimensional zamolodchikov model is derived this hamiltonian is defined on a lattice ofn m sites the special casesn 2 2 m and 3 m are studied

explicit free parametrization of the modified tetrahedron equation the modified tetrahedron equation mte with affine weyl quantum variables at the nth root of unity is solved by a rational mapping operator which is obtained from the solution of a linear problem we show that the solutions can be parametrized in terms of eight free parameters and 16 discrete phase choices thus providing a broad starting point for the construction of three dimensional integrable lattice models the fermat curve points parametrizing the representation of the mapping operator in terms of cyclic functions are expressed in terms of the independent parameters an explicit formula for the density factor of the mte is derived for the example n 2 we write the mte in full detail

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

coxeter transformation and inverses of cartan matrices for coalgebras abstract let c be a coalgebra and let z i c z i c z i c be the grothendieck groups of the category c o p inj and c inj of the socle finite injective right and left c comodules respectively one of the main aims of the paper is to study the coxeter transformation c z i c z i c and its dual c z i c z i c of a pointed sharp euler coalgebra c and to relate the action of c and c on a class of indecomposable finitely cogenerated c comodules n with the ends of almost split sequences starting with n or ending at n by applying chin kleiner and quinn 2002 5 we also show that if c is a pointed k coalgebra such that the every vertex of the left gabriel quiver q c of c has only finitely many neighbours then for any indecomposable non projective left c comodule n of finite k dimension there exists a unique almost split sequence 0 c n n n 0 in the category c comod f c of finitely cogenerated left c comodules with an indecomposable comodule c n we show that dim c n c dim n if c is hereditary or more generally if inj dim d n 1 and hom c c d n 0

knot invariants associated with a particular n continuous limit of the baxter bazhanov model first we briefly recall the definition of the three dimensional baxter bazhanov lattice model the spins of this model are elements ofzn and ther matrix is associated to the algebrauqsl n ifq is a primitiventh root of unity then we construct a particularn limit of the model in which it is meaningful to interpret the spins as elements ofr and which gives the free gaussian boson model finally we study special limits of the rapidity variables in which we obtain braid group representations and we show that forn odd the associated knot invariants are given by the inverse of products of alexander polynomials evaluated at certain roots of unity

labelling schemes for tetrahedron equations and dualities between them zamolodchikov s tetrahedron equations which were derived by considering the scattering of straight strings can be written in three different labelling schemes one can use as labels the states of the vacua between the strings the states of the string segments or the states of the particles at the intersections of the strings we give a detailed derivation of the three corresponding tetrahedron equations and also show how the frenkel moore equations fits in as a non local string labelling we then discuss how an analogue of the wu kadanoff duality can be defined between each pair of the above three labelling schemes it turns out that there are two cases for which one can simultaneously construct a duality between all three pairs of labellings

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

a labelling scheme for higher dimensional simplex equations we present a succinct way of obtaining all possible higher dimensional generalizations of the quantum yang baxter equation qybe using the scheme we could generate the two popular three simplex equations namely the zamolodchikov tetrahedron equation and the frenkel and moore equation

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

corner transfer matrices of the chiral potts model ii the triangular lattice we consider a two dimensional edge interaction model satisfying the star triangle relations for the triangular lattice the corner transfer matrices are functions of three rapidities we show that they possess various factorization properties and satisfy certain equations we indicate how these equations can be solved for the ising model we then consider the three state chiral potts model and obtain low temperature solutions to the equations the conjectured formula for the order parameter the spontaneous magnetization is verified to one more order in a series expansion

inhomogeneous dirichlet boundary value problem for nonlinear schrodinger equations with a power nonlinearity on the upper half plane abstract we consider the inhomogeneous dirichlet boundary value problem for nonlinear schrodinger equations with a power nonlinearity in the upper half plane we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the problem by using the classical energy method and factorization techniques

the emerging role of number theory in exactly solvable models in lattice statistical mechanics abstract we review the baxter model the hard hexagon model and their multistate generalizations from a point of view that stresses the connection with modular functions and additive number theory it is shown for example that various physical quantities in the hard hexagon model are all expressible in terms of modular functions the use of rogers ramanujan type identities in solvable models is also reviewed

tetrahedron equation and generalized quantum groups we construct families of solutions of the yang baxter equation from n products of three dimensional r and l operators satisfying the tetrahedron equation they are identified with the quantum r matrices for the hopf algebras known as generalized quantum groups depending on the number of r s and l s involved in the product the trace construction interpolates the symmetric tensor representations of and the antisymmetric tensor representations of whereas a boundary vector construction interpolates the q oscillator representation of and the spin representation of the intermediate cases are associated with an affinization of quantum superalgebras

is the zamolodchikov model critical evidence is presented in favour of the hypothesis that the zamolodchikov model an exactly solvable three dimensional lattice model without any temperature like parameters is critical the evidence is obtained by generalising the zamolodchikov model to include a temperature like variable the magnetisation curve of this model is then studied using a modified form of a variational approximation formulated earlier also the authors show the two layer zamolodchikov model corresponds to a critical free fermion model

global existence of small amplitude solutions to nonlinear klein gordon equations in four space time dimensions consider the following class of nonlinear wave equations square text u u text text f u u u n k g where t 2 1 2 2 2 3 2 is the d alembertian of the 4 dimensional minkowski space time and f a smooth function of u u t x and its first and second partial derivates vanishing together with its first derivatives at u u u 0 in other words f is at least quadratic in u u u in a neighborhood of the trivial solution u 0 we subject u to the pure initial value problem text u mkern 1mu varepsilon text f x u t varepsilon g x at t 0 i v p with f g c 0 3 and e 0 a small parameter

hom computable coalgebras a composition factors matrix and the euler bilinear form of an euler coalgebra abstract a composition factors matrix f c is studied for any basic hom computable k coalgebra c over an arbitrary field k in connection with a cartan matrix f c of c left euler k coalgebras c are defined they are studied by means of an euler integral bilinear form b c k 0 c k 0 c z the euler characteristic c m n of euler pairs of c comodules m and n and an euler defect c k 0 c k 0 c z of c it is shown that b c lgth m lgth n c m n c m n for all m n in c comod and c 0 if all simple c comodules are of finite injective dimension a diagrammatic characterisation of representation directed hereditary hom computable coalgebras is given

rational mappings arborescent iterations and the symmetries of integrability we describe a class of nonlinear birational representations of groups generated by a finite number of involutions these groups are symmetries of the yang baxter equations and their higher dimensional generalizations they provide discrete dynamical systems with a variety of behaviors from chaotic to integrable according to the number of invariants of the representation

zamolodchikov s tetrahedron equation and hidden structure of quantum groups the tetrahedron equation is a three dimensional generalization of the yang baxter equation its solutions define integrable three dimensional lattice models of statistical mechanics and quantum field theory their integrability is not related to the size of the lattice therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices obviously any such three dimensional model can be viewed as a two dimensional integrable model on a square lattice where the additional third dimension is treated as an internal degree of freedom therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this hidden third dimension in this paper we construct a new solution of the tetrahedron equation which provides in this way the two dimensional solvable models related to finite dimensional highest weight representations for all quantum affine algebra where the rank n coincides with the size of the hidden dimension these models are related to an anisotropic deformation of the sl n invariant heisenberg magnets they were extensively studied for a long time but the hidden 3d structure was hitherto unknown our results lead to a remarkable exact rank size duality relation for the nested bethe ansatz solution for these models note also that the above solution of the tetrahedron equation arises in the quantization of the resonant three wave scattering model which is a well known integrable classical system in 2 1 dimensions

the positive orthogonal grassmannian and loop amplitudes of abjm in this paper we study the combinatorics associated with the positive orthog onal grassmannian ogk and its connection to abjm scattering amplitudes we present a canonical embedding of ogk into the grassmannian gr k 2k from which we deduce the canonical volume form that is invariant under equivalence moves remarkably the canon ical forms of all reducible graphs can be converted into irreducible ones with products of d log forms unlike n 4 super yang mills here the jacobian plays a crucial role to ensure the d log form of the reduced representation furthermore in the positive region we identify the functional map that arises from the triangle equivalence move as a 3 string scattering s matrix which satises the tetrahedron equations by zamolodchikov implying 2 1 dimensional integrability we study the solution to the bcfw recursion relation for loop amplitudes and demonstrate the presence of all physical singularities as well as the absence of all spurious ones the on shell diagram solution to the loop recursion relation exhibits manifest two site cyclic symmetry and reveals that to all loop four and six point amplitudes only have logarithmic singularities

necessary versus sufficient conditions for exact solubility of statistical models on lattices it is shown that under rather mild conditions the triangle relation represents a necessary condition for the existence of commuting transfer matrices of arbitrary size the cases of spin models and vertex models are treated separately

some constant solutions to zamolodchikov s tetrahedron equations the author presents constant solutions to the tetrahedron equations proposed by zamolodchikov 1981 in general from a given solution of the yang baxter equation there are two ways to construct solutions to the tetrahedron equation there are also other kinds of solutions they present some two dimensional solutions that were obtained by directly solving the equations using either an upper triangular or zamolodchikov s ansatz

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

tetrahedral zamolodchikov algebras corresponding to baxter s l operators tetrahedral zamolodchikov algebras are structures that occupy an intermediate place between the solutions of the yang baxter equation and its generalization onto 3 dimensional mathematical physics the tetrahedron equation these algebras produce solutions to the tetrahedron equation and besides specific two layer solutions to the yang baxter equation here the tetrahedral zamolodchikov algebras are studied that arise froml operators of the free fermion case of baxter s eight vertex model

three dimensional vertex model in statistical mechanics from baxter bazhanov model we find that the boltzmann weight of the three dimensional baxter bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the wu kadanoff wegner duality between the cube and vertex type tetrahedron equations is obtained explicitly for the baxter bazhanov model then a three dimensional vertex model is obtained by considering the symmetry property of the weight function which corresponds to the three dimensional baxter bazhanov model the vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube we write down the symmetry relations of the weight functions under the actions of the symmetry groupg of the cube the six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected with the six spaces in which the vertextype tetrahedron equation is defined

bethe ansatz for the three layer zamolodchikov model this paper is a continuation of our previous work boos h e and mangazeev v v 1999 j phys a math gen 32 3041 54 we obtain two more functional relations for the eigenvalues of the transfer matrices for the sl 3 chiral potts model at q2 1 this model up to a modification of boundary conditions is equivalent to the three layer three dimensional zamolodchikov model from these relations we derive the bethe ansatz equations

global existence of small solutions for quadratic quasilinear klein gordon systems in two space dimensions consider a quasi linear system of two klein gordon equations with masses m1 m2 we prove that when m1a2m2 and m2a2m1 such a system has global solutions for small smooth compactly supported cauchy data this extends a result proved by sunagawa j differential equations 192 2 2003 308 in the semi linear case moreover we show that global existence holds true also when m1 2m2 and a convenient null condition is satisfied by the nonlinearities

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

lattice gases and exactly solvable models we detail the construction of a family of lattice gas automata based on a model of t hooft proceeding by use of symmetry principles to define first the kinematics of the model and then the dynamics a spurious conserved quantity appears we use it to effect a radical transformation of the model into one whose spacetime configurations are equivalent to the two dimensional states of an exactly solvable statistical mechanics model the symmetric eight vertex model with parameters restricted to a disorder variety we comment on the implications of this identification for the original lattice gas

new solvable lattice models in three dimensions in this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models the first is the zamolodchikov model which is the only nontrivial model on a three dimen sional lattice so far solved the second is the chiral potts model on the square lattice and its generalization associated with theu q sl n algebra which is of current interest due to its connections with high genus algebraic curves and with representations of quantum groups at roots of unity we show that this last sl n generalized chiral potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square lattice layers with ann valued n 2 spin at each site further in then 2 case this three dimen sional model reduces after a modification of the boundary conditions to the zamolodchikov model we mentioned above

almost split sequences for comodules abstract let be a coalgebra over a field k we introduce an operator tr that takes a right quasi finitely copresented comodule m to a left quasi finitely copresented comodule tr m if m is indecomposable not injective and tr m is finite dimensional over k we prove the existence of an almost split sequence 0 m e dtr m 0 in the category of all right comodules where d homk k if is right semiperfect and the embedding of each simple right comodule s into its injective envelope i s has the property that the socle of i s s is finite dimensional the above almost split sequence exists for each finite dimensional m and dtr m is also finite dimensional

partition function of a three dimensional solvable model abstract the solvable generalized chiral potts model can be interpreted as a three dimensional lattice model with local interactions to within a minor modification of the boundary conditions it is an ising type model with the specific two and three spin interactions on the body centered cubic lattice here we report the results of a calculation of the partition function per site for the infinite lattice

neumann problem for the nonlinear klein gordon equation abstract the neumann initial boundary value problem for the nonlinear klein gordon equation 0 1 v t t v v x x v 3 t x r r v 0 x v 0 x v t 0 x v 1 x x r x v t 0 h t t r for real v 0 x v 1 x and h t is considered we prove the global well posedness for the initial boundary value problem 0 1 and we present a sharp time decay estimate of the solution in the uniform norm also we study the asymptotic behavior of the solution to 0 1 we show that the cubic nonlinearity in the neumann initial boundary value problem 0 1 is scattering critical

an integral arising from the chiral sl n potts model we show that the integral can be expressed in terms of 5f4 hypergeometric functions the integral arises in the solution by baxter and bazhanov of the free energy of the sl n potts model which includes the term j 2 our result immediately gives the logarithmic mahler measure of the laurent polynomial in terms of the same hypergeometric functions