existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in banach spaces in this paper we study nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings in a banach space we prove that ifsis an amenable semigroup and s tt t s is a nonexpansive semigroup on a closed convex subsetcin a uniformly convex banach spaceesuch that the setf s of common fixed points of s is nonempty then there exists a nonexpansive retractionpfromcontof s such thatptt ttp pfor eacht sandpx co ttx t s for eachx c in this case there exists a net a of finite averages of s such that for eacht sand for each bounded subsetbofc lim a ttx a x 0 and lim tta x a x 0 uniformly forx b also if the norm ofeis frechet differentiable then for eachx c pxis the unique common fixed point in s s co ttsx t s furthermore if is an asymptotically invariant net of means then for eachx c t x converges weakly topx

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

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

nonlinear ergodic theorems of nonexpansive type mappings abstract let s be a semitopological semigroup let c be a closed convex subset of a uniformly convex banach space e whose norm is frechet differentiable and t t t s be a continuous representation of s as almost asymptotically nonexpansive type mapping of c into c such that the common fixed point set f of in c is nonempty in this paper we prove that if s is right reversible then for each x c the closed convex set s s c o t t x t s f consists of at most one point we also prove that if s is reversible then the intersection s s c o t t x t s f is nonempty for each x c if and only if there exists a nonexpansive retraction p of c onto f such that p t t t t p p for all t s and px is in the closed convex hull of t t x t s for each x c

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

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 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 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

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

solution of the time dependent liouville von neumann equation dissipative evolution a mathematical and numerical framework has been worked out to represent the density operator in phase space and to propagate it in time under dissipative conditions the representation of the density operator is based on the fourier pseudospectral method which allows a description both in configuration as well as in momentum space a new propagation scheme which treats the complex eigenvalue structure of the dissipative liouville superoperator has been developed the framework has been designed to incorporate modern computer architecture such as parallelism and vectorization comparing the results to closed form solutions exponentially fast convergence characteristics in phase space as well as in the time propagation is demonstrated as an example of its usefulness the new method has been successfully applied to dissipation under the constraint of selection rules more specifically a harmonic oscillator which relaxes to equilibrium under the constraint of second order coupling to the bath was studied

diagonalization of the transfer matrix of a nonintersecting string model abstract in this paper we obtain the eigenvectors of the transfer matrix of a nonintersecting string model using the bethe ansatz the result is in agreement with baxter s conjecture of weak equivalence of the present model with the six vertex model

analyzing the positivity preservation of numerical methods for the liouville von neumann equation abstract the density matrix is a widely used tool in quantum mechanics in order to determine its evolution with respect to time the liouville von neumann equation must be solved however analytic solutions of this differential equation exist only for simple cases additionally if the equation is coupled to maxwell s equations to model light matter interaction the resulting equation set the maxwell bloch or maxwell liouville von neumann mln equations becomes nonlinear in these advanced cases numerical methods are required since the density matrix has certain mathematical properties the numerical methods applied should be designed to preserve those properties we establish the criterion that only methods that have a completely positive trace preserving cptp update map can be used in long term simulations subsequently we assess the three most widely used methods the matrix exponential me method the runge kutta rk method and the predictor corrector pc approach whether they provide this feature and demonstrate that only the update step of the matrix exponential method is a cptp map

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

nonlinear ergodic theorems for general curves defined on general semigroups let g be a algebra semigroup and let h be a real hilbert space let f g be a set of all mappings from g into h in this paper we first try to provide a nonlinear ergodic retraction theorem for the mappings from g to h moreover we introduce the notion of asymptotically nonexpansive type curves and give the ergodic convergence theorems for this kind of curves using the main results we can also prove the ergodic convergence theorem for the almost orbits of semitopological semigroups of non lipschitzian mappings without convexity

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

robust procedures for converting among lindblad kraus and matrix representations of quantum dynamical semigroups given a quantum dynamical semigroup expressed as an exponential superoperator acting on a space of n dimensional density operators eigenvalue methods are presented by which canonical kraus and lindblad operator sum representations can be computed these methods provide a mathematical basis on which to develop novel algorithms for quantum process tomography the statistical estimation of superoperators and their generators from a wide variety of experimental data theoretical arguments and numerical simulations are presented which imply that these algorithms will be quite robust in the presence of random errors in the data

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

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

a solvable lattice model and related rogers ramanujan type identities an exactly solvable lattice model is presented which is a natural generalization of baxter s hard hexagon model to three spin states the free energy and spin densities are calculated related rogers ramanujan type identities are also studied

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

weak convergence and non linear ergodic theorems for reversible semigroups of non lipschitzian mappings abstract letgbe a semitopological semigroup letcbe a closed convex subset of a uniformly convex banach spaceewith afrechetdifferentiable norm and t tt t g be a continuous representation ofgas asymptotically nonexpansive type mappings ofcinto itself such that the common fixed point setf t of t incis nonempty we prove in this paper that ifgis right reversible then for every almost orbitu of t formula consists of at most one point further formula is nonempty for eachx cif and only if there exists a nonexpansive retractionpofcontof t such thatpts tsp pfor alls gandp x is in the closed convex hull of tsx s g x c this result is applied to study the problem of weak convergence of the net u t t g to a common fixed point of t

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

asymptotic behavior of uniformly asymptotically almost nonexpansive curves in a hilbert space abstract we extend the notion of uniformly asymptotically almost nonexpansive curves which was first introduced in rouhani and kim j nonlinear convex anal 4 2003 175 for a commutative semigroup to a general semitopological semigroup study the asymptotic behavior of such curves in a hilbert space and apply our results to study the asymptotic behavior for almost orbits of semigroups of asymptotically nonexpansive type mappings over a general semitopological semigroup in a hilbert space since in this case the semigroup is not directed the proofs in rouhani and kim j nonlinear convex anal 4 2003 175 do not extend to this case and new methods have to be used for the proofs in addition to rouhani j math anal appl 147 1990 465 j math anal appl 151 1990 226 nonlinear anal 44 2001 627 nonlinear anal 51 2002 735 and rouhani and kim j nonlinear convex anal 4 2003 175 our results extend and unify many previously known results j math anal appl 127 1987 206 proc amer math soc 104 1988 431 nonlinear anal 29 1997 539 topol methods nonlinear anal 5 1995 305 j math anal appl 105 1985 514 nonlinear anal 26 1996 1411 pacific j math 126 1987 277 j math anal appl 206 1997 451 chinese ann math 6 1996 729 nonlinear anal 14 1990 69 j math anal appl 85 1982 172 proc amer math soc 81 1981 253 proc amer math soc 97 1986 55 canad j math 44 1992 880 j math anal appl 142 1989 242 proc amer math soc 117 1993 385

computing the action of the matrix exponential with an application to exponential integrators a new algorithm is developed for computing e ta b where a is an n times n matrix and b is n times n 0 with n 0 ll n the algorithm works for any a its computational cost is dominated by the formation of products of a with n times n 0 matrices and the only input parameter is a backward error tolerance the algorithm can return a single matrix e ta b or a sequence e t ka b on an equally spaced grid of points t k it uses the scaling part of the scaling and squaring method together with a truncated taylor series approximation to the exponential it determines the amount of scaling and the taylor degree using the recent analysis of al mohy and higham siam j matrix anal appl 31 2009 pp 970 989 which provides sharp truncation error bounds expressed in terms of the quantities a k 1 k for a few values of k where the norms are estimated using a matrix norm estimator shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems and analysis of rounding errors and of the conditioning of the problem provides theoretical support experimental comparisons with matlab codes based on krylov subspace chebyshev polynomial and laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy an important application of the algorithm is to exponential integrators for ordinary differential equations it is shown that the sums of the form sum k 0 p varphi k a u k that arise in exponential integrators where the varphi k are related to the exponential function can be expressed in terms of a single exponential of a matrix of dimension n p built by augmenting a with additional rows and columns and the algorithm of this paper can therefore be employed

asymptotic behavior of almost orbits of reversible semigroups of lipschitzian mappings abstract let c be a nonempty closed convex subset of a hilbert space h g a right reversible semitopological semigroup and s s t t g a continuous representation of g as lipschitzian mappings of c into itself then we deal with the asymptotic behavior of an almost orbit u t t g of s s t t g we first prove that if limsupt kt 1 then the closed convex subset s e g co u t t s f s consists of at most one point where kt is the lipschitzian constant of s t t g then we apply this result to study the problem of weak convergence of the net u t t g

time domain simulations of the nonlinear maxwell equations using operator exponential methods in this paper we propose a krylov subspace based operator exponential method for time domain simulations of the maxwell equations with general nonlinear polarizations this includes classical chi 2 or chi 3 nonlinearities and or nonlinear coupled system dynamics as an illustration we compare the performance of our approach to certain well known methods for the case of pulse self steepening in a material with negative kerr nonlinearity for this system we also develop an appropriate analytical reference solution in addition we demonstrate that our approach allows to treat the complex nonlinear dynamics of various physical systems classical and or quantum coupled to electromagnetic fields