algebraic bethe ansatz for the gl 1 2 generalized model ii the three gradings the algebraic bethe ansatz can be performed rather abstractly for whole classes of models sharing the same r matrix the only prerequisite being the existence of an appropriate pseudo vacuum state here we perform the algebraic bethe ansatz for all models with 9 9 rational gl 1 2 invariant r matrix and all three possibilities of choosing the grading our bethe ansatz solution applies for instance to the supersymmetric t j model the supersymmetric u model and a number of interesting impurity models it may be extended to obtain the quantum transfer matrix spectrum for this class of models the properties of a specific model enter the bethe ansatz solution i e the expression for the transfer matrix eigenvalue and the bethe ansatz equations through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model

pseudoholomorphic strips in symplectizations ii fredholm theory and transversality this paper is part of a larger program the investigation of the chord problem in three dimensional contact geometry the main tool will be pseudoholomorphic strips in the symplectisation of a three dimensional contact manifold with two totally real submanifolds l0 l1 as boundary conditions the submanifolds l0 and l1 do not intersect transversally in this paper we will develop a nonlinear fredholm theory that guarantees the existence of a family of embedded pseudoholomorphic strips near a given one with suitable properties c 2003 wiley periodicals inc

exotic quantum double its universal r matrix and their representations the exotic quantum double and its universal r matrix for the quantum yang baxter equation are constructed in terms of drinfeld s quantum double theory as a new quasitriangular hopf algebra it is different from those standard quantum doubles that are the q deformations for lie algebras or lie superalgebras by studying its representation theory many parameter representations of the exotic quantum double are obtained with an explicit example associated with lie algebra a2 the multiparameter r matrices for the quantum yang baxter equation can result from the universal r matrix of this exotic quantum double and these representations

the dynamics on three dimensional strictly convex energy surfaces we show that a hamiltonian flow on a three dimensional strictly convex energy surface s c r4 possesses a global surface of section of disc type it follows in particular that the number of its periodic orbits is either 2 or oc by a recent result of j franks on area preserving homeomorphisms of an open annulus in the plane the construction of this surface of section is based on partial differential equations of cauchy riemann type for maps from punctured riemann surfaces into r x s3 equipped with special almost complex structures

weight functions and drinfeld currents a universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its borel subalgebra satisfying certain coalgebraic properties in representations of the quantum affine algebra it gives off shell bethe vectors and is used in the construction of solutions of the qkz equations we construct a universal weight function for each untwisted quantum affine algebra using projections onto the intersection of borel subalgebras of different types and study its functional properties

a new exactly solvable case of an o n model on a hexagonal lattice a new solvable case of o n model on a hexagonal lattice is presented it is shown that the model has a phase transition at n 2 this phase transition can be interpreted as phase transition between percolating and non percolating phases

classical yang baxter equations and quantum integrable systems quantum integrable models associated with nondegenerate solutions of classical yang baxter equations related to the simple lie algebras are investigated these models are diagonalized for rational and trigonometric solutions in the cases of sl n gl n o n and sp n algebras the analogy with the quantum inverse scattering method is demonstrated

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

scalar products of bethe vectors in models with mathfrak gl 2 1 symmetry 1 super analog of reshetikhin formula we study scalar products of bethe vectors in integrable models solvable by nested algebraic bethe ansatz and possessing mathfrak gl 2 1 symmetry using explicit formulas of the monodromy matrix entries multiple actions onto bethe vectors we obtain a representation for the scalar product in the most general case this explicit representation appears to be a sum over partitions of the bethe parameters it can be used for the analysis of scalar products involving on shell bethe vectors as a by product we obtain a determinant representation for the scalar products of generic bethe vectors in integrable models with mathfrak gl 1 1 symmetry

quantum integrability and quantum groups abstract this report presents a brief summary of the theory of quantum integrable systems first the quantum inverse scattering method is used to give an algebraic formulation of the bethe ansatz method second the yang baxter relation a sufficient condition for integrability is introduced and its relation to quantum groups is discussed third some recent results on quantum integrable systems with long range interactions are explained to present the basic ideas and concepts in an introductory way specific models are chosen the results however apply to a wider class of quantum integrable models

fast analytic solver of rational bethe equations in this note we propose an approach for a fast analytic determination of all possible sets of bethe roots corresponding to eigenstates of rational integrable spin chains of given not too large length in terms of baxter q functions we observe that all exceptional solutions if any are automatically correctly accounted the key intuition behind the approach is that the equations on the q functions are determined solely by the young diagram and not by the choice of the rank of the symmetry hence we can choose arbitrary and that accommodate the desired representation then we consider all distinguished q functions at once not only those following a certain kac dynkin path

o n matrix difference equations and a nested bethe ansatz a system of u n matrix difference equations is solved by means of a nested version of a generalized bethe ansatz the highest weight property of the solutions is proved and some examples of solutions are calculated explicitly

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

non abelian su 3 _ k anyons inversion identities for higher rank face models the spectral problem for an integrable system of particles satisfying the fusion rules of is expressed in terms of exact inversion identities satisfied by the commuting transfer matrices of the integrable fused interaction round a face model of jimbo miwa and okado the identities are proven using local properties of the boltzmann weights in particular the yang baxter equation and unitarity they are closely related to the consistency conditions for the construction of eigenvalues obtained in the separation of variables approach to integrable vertex models

a non standard quantum double as the l state boson algebra and its r matrices for the quantum yang baxter equation in this paper we construct a new quantum double by endowing the l state boson algebra with a non trivial hopf algebra structure which is not a q deformation of the lie algebra or superalgebra the universal r matrix for the yang baxter equation associated with this new quantum group structure is obtained explicitly by building the representations of this quantum double we obtain some r matrices which can result in new representations of the braid group

off shell bethe ansatz equations and n point correlators in the su 2 wznw theory we prove that the wavevectors of the off shell bethe ansatz equation for the inhomogeneous su 2 lattice vertex model render in the quasiclassical limit the solutions of the knizhnik zamolodchikov equation

algebraic bethe ansatz for the fpl2 model an exact solution of the model of fully packed loops of two colours on a square lattice has recently been proposed by dei cont and nienhuis using the coordinate bethe ansatz approach we point out here a simpler alternative in which the transfer matrix is directly identified as a product of r matrices this allows us to apply the nested algebraic bethe ansatz which leads to the same bethe equations we comment on some of the applications of this result

limit cycle generation for a class of non linear systems with jumps using a low dimensional predictive control in this paper it is shown that using a low dimensional non linear predictive control scheme provably stable limit cycles can be obtained for open loop non linear systems with unstable equilibrium point a particular case where the limit cycle may be reduced to a single point in the state space can be obtained which corresponds to asymptotic stabilization the system may present hybrid nature in the sense that discontinuities jump phenomena on the state evolution may be handled the proposed feedback scheme holds for classical jump free systems as a particular case the proposed strategy is illustrated through two examples a jump free system the ball and beam and a non linear hybrid dynamical system including state jumps the modified impulsive lorenz chaotic system

diagonalisation of gl n invariant transfer matrices and quantum n wave system lee model the algebraic bethe ansatz is constructed for the gl n invariant transfer matrices with arbitrary gl n spin for this purpose the notion of vacuum subspace is introduced it is shown that the gl n magnet can be interpreted as an integrable discrete approximation of vector or matrix nonlinear schrodinger models or of the quantum n wave system

generalized n level jaynes cummings and dicke models classical rational r matrices and algebraic bethe ansatz we construct integrable n level generalizations of jaynes cummings and dicke models corresponding to the simple reductive lie algebras of rank n we show that for each such lie algebra there exist many integrable jaynes cummings and dicke type models each of which is associated with the reductive subalgebras containing cartan subalgebra and is obtained via reduction from jaynes cummings or dicke type models with a maximal number of bosons we diagonalize the constructed jaynes cummings and dicke type hamiltonians in the physically most interesting case of using a nested algebraic bethe ansatz

the bcs model and the off shell bethe ansatz for vertex models we study the connection between the bcs pairing model and the inhomogeneous vertex model the two spectral problems coincide in the quasi classical limit of the off shell bethe ansatz of the disordered six vertex model the latter problem is transformed into an auxiliary spectral problem which corresponds to the diagonalization of the integrals of motion of the bcs model a generating functional whose quasi classical expansion leads to the constants of motion of the bcs model and in particular the hamiltonian is identified