a family of methods for the solution of lattice models abstract we discuss a family of methods for evaluating the thermodynamic functions for a class of lattice models that includes the ising model it is shown that the usual transfer matrix method can be considered as a member of the family some typical members of this family are also introduced furthermore we are concerned with the acceleration of these methods for this reason an extra linear extrapolation algorithm which is based on the member of this family related to the transfer matrix method is presented by this algorithm not only can the bulk physical quantities be evaluated more precisely but also an approximation to the boundary thermodynamic quantities can be obtained a new expression for the boundary free energy of the ising model is derived in addition the convergence to the thermodynamic limit is proven for the scaling method presented by chorin some numerical results for the bulk and boundary thermodynamic functions of the ising model are included

perturbation expansions of the axial next nearest neighbour ising and asymmetric four state clock models the axial next nearest neighbour ising annni and four state asymmetric clock 4 asc models are investigated by perturbing about points at which they reduce the simple nearest neighbour ising models these limits are a for the annni model i small nearest neighbour interaction j1 in the x direction and ii small next nearest neighbour interaction j2 in the x direction and b for the 4 asc model small asymmetry delta in the limit j1 to 0 the phase boundary is located to first order in j1 and found to be ising like in the limit j2 to 0 the behaviour to second order in j2 is more complicated but is identical to that found to second order in delta for the 4 asc model these results are discussed in the light of other recent work

modules of highest weight type for the algebra of derivations over a rational quantum torus for the lie algebra of derivations over a rational quantum torus we construct a class of irreducible weight modules with all finite dimensional weight spaces which are called the modules of highest weight type generating the notion for the case of commutative torus for the lie algebra of derivations over a rational quantum torus we construct a class of irreducible weight modules with all finite dimensional weight spaces which are called the modules of highest weight type generating the notion for the case of commutative torus

integrable representations of toroidal lie algebras co ordinatized by rational quantum tori for any positive integers d n with d 1 and n 1 we fix an n by n complex matrix q qij satisfying qii 1 and qij qji 1 with all qij roots of unity let d q be the universal central extension of the lie subalgebra sld cq of gld cq with trace in cq cq where cq is the rational quantum torus associated to q and let d q be the lie algebra by adding the n degree derivations to d q with respect to the n non commuting variables in cq the lie algebra d q called the toroidal lie algebra co ordinatized by the rational quantum torus cq has an n dimensional center c in this paper we obtain a classification of irreducible integrable modules with finite dimensional weight spaces and with non zero center action over the toroidal lie algebra d q

conformal invariance and the critical behavior of the triplet xy quantum chain we introduce and study the critical properties of the tripletxy quantum chain this system is described in terms of three spin interactions and is the generalization of the standardxy quantum chain we show that this model with periodic boundaries has a local gauge invariance and can be described by the composition of two triplet ising models with general toroidal boundary conditions from this composition the phase diagram as well the conformai anomaly and critical exponents are determined by exploring their relations with the mass gap amplitudes predicted by conformai invariance

convergence rates for the approximations of the solutions to algebraic riccati equations with unbounded coefficients case of analytic semigroups approximations schemes for the solutions of the algebraic riccati equations will be considered we shall concentrate on the case when the input operator is unbounded and the dynamics of the system is described by an analytic semigroup the main goal of the paper is to establish the optimal rates of convergence of the underlying approximations by optimal we mean such approximations which would reconstruct the optimal regularity of the original solutions as well as the best approximation properties of the finite dimensional subspaces it turns out that if one aims to obtain the optimal rates in the case of unbounded input operators the choice of the approximations to the generator as well as to the control operator is very critical while the convergence results hold with any consistent approximations the optimal rates require a careful selection of the approximating schemes our theoretical results will be illustrated by several examples of boundary point control problems where the optimal rates of convergence are achieved with the appropriate numerical algorithms

structure of a new class of non graded infinite dimensional simple lie algebras abstract a new class of infinite dimensional simple lie algebras over a field with characteristic 0 are constructed these are examples of non graded lie algebras the isomorphism classes of these lie algebras are determined the structure space of these algebras is given explicitly

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

finite dimensional irreducible representations of elementary unitary lie algebras over quantum tori abstract we classify all finite dimensional irreducible representations of the elementary unitary lie algebras eu n c q over the quantum torus with anti involution c q for n 5

conformal invariance and the operator content of the xxz model with arbitrary spin the authors are concerned with the critical properties of the anisotropic heisenberg chain or xxz model with arbitrary integer or half integer spin the eigenspectra of these hamiltonians with periodic boundaries are calculated for finite chains by solving numerically their associated bethe ansatz equations the resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified for lattices with an even and odd number of sites the results for spin 1 and spin 3 2 indicate that the conformal anomaly c for these models in the gapless regime has the value c 3s 1 s independent of the anisotropy and the exponents vary continuously with the anisotropy as in the eight vertex model the operator content of these models indicate that the underlying field theory governing these critical spin s models is described by composite fields formed by the product of gaussian and z n fields with n 2s finally some of the irrelevant operators which produce the leading finite size corrections of the eigenenergies are identified

the linear regulator problem for parabolic systems we present an approximation framework for computation in finite dimensional spaces of riccati operators that can be guaranteed to converge to the riccati operator in feedback controls for abstract evolution systems in a hilbert space it is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems

deformations of lie algebras using σ derivations in this article we develop an approach to deformations of the witt and virasoro algebras based on sigma derivations we show that sigma twisted jacobi type identity holds for generators of such deformations for the sigma twisted generalization of lie algebras modeled by this construction we develop a theory of central extensions we show that our approach can be used to construct new deformations of lie algebras and their central extensions which in particular include naturally the q deformations of the witt and virasoro algebras associated to q difference operators providing also corresponding q deformed jacobi identities c 2005 elsevier inc all rights reserved

long range one dimensional potts models a cluster mean field and extrapolation approach we obtain approximations for the critical temperature tc of q state ferromagnetic potts models on one dimensional lattices with algebraically decaying ferromagnetic pair interactions i e decaying as 1 r with 1 2 initially we use a cluster mean field method to get approximations to the critical temperature and we look at increasingly large clusters this gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms we see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate tc estimates our best estimates we believe to be correct to four figure accuracy

finite size scaling study of the equilibrium cluster distribution of the two dimensional ising model the authors use a very fast and efficient algorithm to study by monte carlo methods the equilibrium cluster distribution cl l and the mean number of clusters per lattice site containing l particles in a square lattice of l2 sites of the two dimensional ising model at the critical point finite size scaling theory is then used to analyse the scaling ansatz cl l l tau f ls l tau and s being critical exponents the second moment of the cluster distribution p2 l sigma l2cl is shown to behave as p2 l approximately ltheta with theta 1 895 or 0 010 the effect of corrections to scaling is also discussed

a characterization of the block lie algebra and itsq forms in characteristic 0 abstract the block algebra l referred to here is the lie algebra over a field f of characteristic 0 with basis e r s r s z z 0 0 and subject to the commutation relations e h k e r s rk sh e h r k s let 0 1 q f the q form l q of l is the lie algebra with basis e r s q r s z z 0 0 and subject to the commutation relations e h k q e r s q q rk q sh e h r k s q we study here a z z graded lie algebra formula with a 0 0 0 and dim a i j 1 for all i j z z 0 0 in this paper we show that a is necessarily isomorphic to l or l q l q l q for some q f if a satisfies a certain nondegeneracy condition

derivations of skew polynomial rings abstract let f be a field of characteristic 0 and let i j f for 1 i j n define r f x 1 x 2 x n to be the skew polynomial ring with x i x j i j x j x i and let s f x 1 x 2 x n x 11 x 12 x 1n be the corresponding laurent polynomial ring in a recent paper kirkman procesi and small considered these two rings under the assumption that s is simple and showed for example that the lie ring of inner derivations of s is simple furthermore when n 2 they determined the automorphisms of s related its ring of inner derivations to a certain block algebra and proved that every derivation of r is the sum of an inner derivation and a derivation which sends each xi to a scalar multiple of itself in this paper we extended these results to a more general situation specifically we study twisted group algebras ft g where g is a commutative group and f is a field of any characteristic furthermore we consider certain subalgebras ft h where h is a subsemigroup of g which generates g as a group finally if e g g f is a skew symmetric bilinear form then we study the lie algebra fe g associated with e and we consider its relationship to the lie structure defined on various twisted group algebras ft g

classification of irreducible bounded weight modules over the derivation lie algebras of quantum tori abstract let d 1 be an integer q q i j d d a d d complex matrix satisfying q i i 1 q i j q j i 1 with all q i j being roots of unity let c q be the rational quantum torus algebra associated with q and der c q its derivation lie algebra in this paper we give a complete classification of irreducible bounded weight modules over der c q they turn out to be irreducible sub quotients of der c q module v v w for a finite dimensional irreducible gl d module v a finite dimensional graded irreducible gl n module w and c d where the integer n is uniquely determined by q

finite lattice methods in quantum hamiltonian field theory i o 2 and o 3 heisenberg models for pt i see ibid vol 14 p 241 57 1981 two efficient methods for finding the low lying states of hamiltonians on finite lattices are described the first involves constructing a finite representation of the hamiltonian using strong coupling eigenstates while the second is based on the lanczos recursion method the methods are used to determine the mass gap of the o 2 and o 3 heisenberg hamiltonians in 1 1 dimensions for a sequence of finite chains the critical behaviour of the infinite chain is then analysed by extrapolating the finite lattice estimates using finite size scaling a remarkably sensitive test is developed for the presence of a phase transition for the o 2 model data this test yields strong evidence for a phase transition with the weak coupling phase massless while in the o 3 case the test supports although more weakly the absence of any transition

droplet theory in low dimensions ising systems in zero field the authors develop a theory of the universal configurational physics underlying critical point phenomena in the ising universality class the theory is formally justifiable in d 1 epsilon dimensions and may be regarded as the natural continuation of the kink based theory of one dimension which it incorporates as a limiting case in d 1 epsilon the configurational building block is the droplet the typical droplet is not spherical and the many droplet assembly is not dilute the implied problems are handled with renormalisation group methods it is found that droplet shape fluctuation effects control the correlation length exponent nu while the nesting of droplets within droplets control the order parameter exponent beta the exponents nu and beta thus effectively define respectively the fractal dimensions of the droplet surface and the droplet volume the theory is used to determine and illuminate the critical behaviour of further quantities including the free energy the susceptibility the droplet number distribution and the distribution of the intra droplet order

finite size scaling in the 2 1 d ising model hamiltonian eigenvalues are calculated for the 2 1 d ising model on lattices up to 5 5 sites finite size scaling techniques are used to estimate the critical coupling xc 0 329 or 0 001 and index v 0 635 or 0 005 the prospects are discussed for further applications of these methods in three and four dimensions

finite size scaling and conformal invariance in a self dual quantum z 5 model considers the critical behaviour of a particular set of onedimensional self dual models with z n symmetry n or 5 the critical indices are evaluated using standard finite size scaling and by exploiting their relations with the mass gap amplitudes predicted by conformal invariance the results strongly suggest that the recently introduced z n quantum field theory is the underlying field theory for these statistical mechanics models

finite size scaling studies of the o 2 heisenberg spin model in d 1 dimensions the finite size scaling behaviour of the o 2 heisenberg spin model in d 1 dimensions d 1 and 2 is computed in the low temperature region using the stochastic truncation monte carlo method the numerical results so obtained agree very well with effective lagrangian theory and with spin wave perturbation theory

equilibrium cluster distributions of the three dimensional ising model in the one phase region abstract we analyse equilibrium cluster distributions obtained numerically from a ferromagnetic ising model simple cubic lattice 125000 sites and periodic boundary conditions along the coexistence line and in the one phase region below t c we find evidences that the distribution of sizes and energies scales with temperature and external magnetic field giving binder s droplet exponent y 4 9 the mean number of incident interior and exterior bonds on a cluster of size l s l seems to behave as l x with x 9 10 when not far away from t c we conclude that while the classical nucleation theory may provide an approximate description around 0 59 t c it has to be modified at higher and lower temperatures the fisher droplet model and the approach by penrose et al based on a renormalized fugacity are also discussed we thus obtain simple semiphenomenological expressions for the cluster equilibrium distributions and partition functions

convergence and extrapolation in finite size scaling renormalization abstract finite size renormalization technique is studied by examining a test model which is exactly solvable and the problem of lattice animals and self avoiding walks bulk estimates of the leading irrelevant rg eigenvalue are found and compared with independent estimates from finite size data we introduce an extrapolation technique which is applicable to non monotonic sequences of exponent approximants

latent heats from finite size scaling a method of obtaining the latent heat at a first order transition from finite lattice matrix elements is discussed and demonstrated for the case of the 1 1 d potts model the method is shown to provide smoothly convergent finite lattice sequences the structure of the finite lattice hamiltonian eigenvalues is exhibited and a characteristic signal of the first order transition is proposed the finite size scaling behaviour at the transition is also discussed

droplet theory in low dimensions ising systems in an ordering field the authors extend the microscopic droplet theory of ising systems developed recently to incorporate the effects of an ordering field the theory yields a free energy which is the solution of a renormalisation group equation describing droplet nesting and which has a full scaling form the behaviour near the coexistence curve is investigated and found to display the essential singularity suggested by primitive droplet models but with parameters renormalised in a physically intelligible way the droplet population function implied by the theory is parametrised in a simple way in space dimension d 2 to yield predictions for various thermodynamic properties including the equation of state which are found to be in fair accord with series based results the structure of the field dependent droplet number distributions is investigated and contrasted with the forms assumed in phenomenological droplet theories the domain of validity of the theory is assessed it is concluded that the theory fails to describe sufficiently large droplets in the presence of a finite field and is thus in principle trustworthy only in a region close to the coexistence curve

theq virasoro like algebra abstract in this paper finite dimensional quotients of the q virasoro like algebra are determined when q is a root of unity as an application finite dimensional irreducible modules over the q virasoro like algebra are classified for any root q of unity

the mean size of ising clusters a confluent singularity analysis the authors re examine using a confluent singularity analysis existing low temperature series for the mean size of finite clusters for the pure spin 1 2 nearest neighbour ising model in two dimensions although the triangular lattice is problematical results for the honeycomb and square lattices may be interpreted as indicating a dominant exponent theta 1 87 or 0 04 consistent with the theoretical prediction theta gamma beta together with competing analytic and non analytic delta 1 1 35 or 0 25 correction to scaling terms

convergence rates for the feedback operators arising in the linear quadratic regulator problem governed by parabolic equations the linear quadratic control problem for systems governed by evolution equations of parabolic type is considered the optimal feedback control uses a riccati operator which is approximated by a galerkin scheme optimal convergence rates except for a logarithmic factor are derived for the approximation of the riccati operator for both the finite and infinite time interval this is done in an abstract hilbert space framework assuming a certain rate of convergence for the uncontrolled system in the numerical part the galerkin method is compared with the collocation method with special emphasis on the feedback gains