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

elliptic solutions to difference non linear equations and related many body problems we study algebro geometric finite gap and elliptic solutions of fully discretized kp or 2d toda equations in bilinear form they are hirota s difference equation for functions starting from a given algebraic curve we express the function and the baker akhiezer function in terms of the riemann theta function we show that the elliptic solutions when the function is an elliptic polynomial form a subclass of the general algebro geometric solutions we construct the algebraic curves of the elliptic solutions the evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the ruijsenaars schneider many body system the zeros obey equations which have the form of nested bethe ansatz equations known from integrable quantum field theories we discuss the lax representation and the action angle type variables for the many body system we also discuss elliptic solutions to discrete analogues of kdv sine gordon and 1d toda equations and describe the loci of the zeros

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

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

the uniqueness of groups of lyons type we give the first computer free proof of the uniqueness of groups of type ly we also show that certain simpicial complexes associated to the lyons group ly have the same homotopy type as the quillen complex for ly at the prime 3 we show this complex is simply connected and we calculate its homology finally we supply simplified proofs of some properties of groups of type ly such as the group order a group of type ly is a finite group g possessing an involution t such that h cg t is the covering group of the alternating group a1l and t is not weakly closed in h with respect to g we prove

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

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

the mcmillan and newton polygons of a feedback system and the construction of root loci abstract the local behaviour of root loci around zeros and poles is investigated this is done by relating the newton diagrams which arise in the local analysis to the mcmillan structure of the open loop system by means of what we shall call the mcmillan polygon this geometric construct serves to clarify the precise relationship between the mcmillan structure the principal structure and the branching patterns of the root loci in addition several rules are obtained which are useful in the construction of the root loci of multivariable control systems

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

critical behavior of self avoiding walks on fractals using a new graph counting technique suitable for self similar fractals exact 18th order series expansions for saws on some sierpinski carpets are generated from them the critical fugacityxc and critical exponents saw and saw are obtained the results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied we find for some carpets with low lacunarity that saw saw d for saws on the fractals which are embedded in ad dimensional euclidean space

cohomological structure in soliton equations and jacobian varieties on etudie la structure des orbites du systeme dynamique definie par la hierarchie kp totale on montre que chaque orbite est localement isomorphe a un certain groupe de cohomologie associe a une algebre commutative le systeme dynamique kp restreint a chaque orbite determine un systeme dynamique de mouvements lineaires sur lui par rapport a la structure lineaire du groupe de cohomologie

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

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

conditions for scale based decompositions in singularly perturbed systems abstract singularly perturbed models of the form x t a x t with a analytic at 0 nonsingular for 0 0 and singular at 0 arise naturally in various problems of systems and control theory under a so called multiple semisimple null structure mssns condition on a the eigenstructure of this matrix has a multiple scale property that allows the asymptotic eigenstructure of the matrix to be studied via reduced order matrices associated with the separate scales under a stronger multiple semistability or msst condition this eigenstructure decomposition translates into a time scale decomposition of the solution x t of the system this paper is aimed at illuminating the mssns and msst conditions using ideas from an algebraic approach that we have developed for the study of singularly perturbed systems we show among other results that the smith decomposition of a permits transformation to a form in which the scales become explicit and the computations become transparent and simple that this form allows us to identify perturbations of a that all have the same scale based decompositions that the eigenstructure of a does indeed display multiple scales under mssns and that the eigenvalues in this case can be approximated using reduced order calculations that a has mssns if and only if the orders of its invariant factors equal the orders of its eigenvalues that this happens if and only if the orders of its invariant factors and principal minors are related in a specified way and that a has msst if and only if it is hurwitz for 0 0 and has mssns and satisfies the requirement that the order in of the real part of every eigenvalue is not greater than the order of its imaginary part

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

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

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

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

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

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

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