conformal invariance and finite one dimensional quantum chains based on previous work of cardy 1984 the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions the method is illustrated in the case of the three and four state potts models where the anomalous dimensions of the para fermionic operators are found

transmission conditions on interfaces for hamilton jacobi bellman equations we establish a comparison principle for a hamilton jacobi bellman equation more appropriately a system related to an infinite horizon problem in presence of an interface namely a low dimensional subset of the state variable space where discontinuities in controlled dynamics and costs take place since corresponding hamiltonians at least for the subsolution part do not enjoy any semicontinuity property the comparison argument is rather based on a separation principle of the controlled dynamics across the interface for this we essentially use the notion of partition and minimal partition for intervals of definition of an integral trajectory

asymptotic stability for impulsive differential systems employing piecewise continuous lyapunov functions results of stability in terms of two different measures for impulsive differential systems are derived an example is worked out

on the phase diagram of the discrete z6 spin models we point out some problems with the previously proposed phase diagram of the z6 spin models consideration of the diagram near to the decoupling surface using both exact and approximate arguments suggests a modification which remedies these deficiencies with the aid of a new parametrization of the phase space we study the models numerically with results which support our conjectures

infinite horizon problems on stratifiable state constraints sets this paper deals with a state constrained control problem it is well known that unless some compatibility condition between constraints and dynamics holds the value function has not enough regularity or can fail to be the unique constrained viscosity solution of a hamilton jacobi bellman hjb equation here we consider the case of a set of constraints having a stratified structure under this circumstance the interior of this set may be empty or disconnected and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints in such situations the classical pointing qualification hypothesis are not relevant the discontinuous value function is then characterized by means of a system of hjb equations on each stratum that composes the state constraints this result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur

parafermions and surface exponents of self dual z n spin models from conformal invariance the author studies the critical behaviour of a family of self dual two dimensional spin models with z n symmetry n or 8 in its hamiltonian formulation using the relations between the mass gap amplitudes of the hamiltonian in a finite strip and the critical exponents he estimates the anomalous dimensions of the parafermions occurring in the underlying field theory as well as the surface exponents

the ashkin teller quantum chain and conformal invariance the authors study the finite size effects for the fourth state one dimensional quantum chain introduced by kohmoto et al 1981 using different boundary conditions from the corrections to the ground state energy for periodic boundary conditions they verify that the central charge of the virasoro algebra is equal to one in the whole domain of criticality their numerical study suggests a new relation between the corrections to the ground state energy with free boundary conditions and with periodic boundary conditions various critical exponents including the surface ones and those corresponding to the para fermions are determined

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

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

phase diagram in the generalized chiral clock models abstract using the results of the integrable chiral potts model we speculate about the possible phase diagram of the generalized chiral clock model in the integrable model it is found that the chiral fields are almost proportional to one another here we consider a chiral model with all the chiral fields exactly proportional to one another with fixed proportionality constants given by the integrable case this reduces the chiral field variables to just one denoted by the model thus defined includes the generalized clock model or z n model at 0 or 1 2 and the chiral clock models as special cases using the results of many authors on the z n model in the so called symmetric case with vertical and horizontal interactions equal and the behavior of the system at t 0 we can get a pretty good picture of the phase diagram of this chiral model the exact results show that the commonly held belief that the massless phase in the z n model for n 5 is in the same universality class as the classical xy model and the belief that the commensurate incommensurate phase transition is of the pokrovskii talapov type and the incommensurate fluid transition is of kosterlitz thouless type may not be correct as universality may not even hold on these phase boundaries

operator content of n state quantum chains in the c 1 region the authors conjecture the operator content for n state quantum chains n or 5 in the domains of the coupling constants where the central charge of the virasoro algebra is equal to one free boundary conditions as well as boundary conditions compatible with the torus are considered the conjectured operator content is compared with finite size scaling estimates

discontinuous solutions of hamilton jacobi equations on networks this paper studies optimal control problems on networks without controllability assumptions at the junctions the value function associated with the control problem is characterized as solution to a system of hamilton jacobi equations with appropriate junction conditions the novel feature of the result lies in that the controllability conditions are not needed and the characterization remains valid even when the value function is not continuous

multi particle structure in the zn chiral potts models the authors calculate the lowest translationally invariant levels of the z3 and z4 symmetrical chiral potts quantum chains using numerical diagonalization of the hamiltonian for n or 12 and n or 10 sites respectively and extrapolating n to infinity in the high temperature massive phase they find that the pattern of the low lying zero momentum levels can be explained assuming the existence of n 1 particles carrying zn charges q 1 n 1 mass mq and their scattering states in the superintegrable case the masses of the n 1 particles become proportional to their respective charges mq qm1 exponential convergence in n is observed for the single particle gaps while power convergence is seen for the scattering levels they also verify that qualitatively the same pattern appears for the self dual and integrable cases for general zn they show that the energy momentum relations of the particles show a parity nonconservation asymmetry which for very high temperatures is exclusive due to the presence of a macroscopic momentum pm 1 2q n phi where phi is the chiral angle and q is the zn charge of the respective particle

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

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

the spin s xxz quantum chain with general toroidal boundary conditions the authors derive the operator content of the spin s xxz quantum chain with generalised toroidal boundary conditions compatible with the u 1 symmetry of the model these results are derived by solving numerically the associated bethe ansatz equations for finite chains and exploring the consequences of the conformal invariance of the infinite system they also show that as in the spin 1 2 case the conformal anomaly and dimensions of general extended su 2 algebras can be obtained from these spin s chains by choosing properly the coupling constant and the boundary condition

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

conformal invariance and the critical behaviour of a quantum spin hamiltonian with three spin coupling a quantum spin model with three spin coupling is studied by finite lattice methods the critical indices nu and alpha are obtained by standard finite size scaling analysis conformal invariance is used to estimate the exponents nu and eta the central charge or conformal anomaly c and the anomalous dimension of a spin 1 4 parafermion the authors conclude that the model belongs to the same universality class as the four state potts and baxter wu models although the convergence of the various finite lattice estimators is extremely slow

a new formulation of state constraint problems for first order pdes the first order hamilton jacobi bellman equation associated with the state constraint problem for optimal control is studied instead of the boundary condition which soner introduced a new and appropriate boundary condition for the pde is proposed the uniqueness and lipschitz continuity of viscosity solutions for the boundary value problem are obtained

z 6 model criticality and break collapse method abstract the phase diagram and thermal and crossover exponents of the z 6 ferromagnet on the square lattice are calculated within a real space renormalization group rg scheme the obtained phase diagram exhibiting 4 phases corresponding to z 6 z 3 z 2 and completely broken symmetries contains all exactly known critical points and possibly is an excellent approximation everywhere except in the high temperature region where a soft phase is expected to appear however all the obtained results are exact in the wheatstone bridge hierarchical lattice in addition to these results we present an operational procedure the break collapse method which considerably simplifies the exact calculation of two spin correlation functions for arbitrary z 6 two rooted clusters frequent in rg approaches

the bond diluted z q ferromagnetic model criticality and break collapse method within a real space renormalization group rg which preserves two site correlation functions we study on the square lattice the criticality of the bond diluted z q ferromagnetic model we generalize the break collapse method which simplifies greatly the exact calculation of arbitrary z q two terminal clusters commonly appearing in rg approaches mainly for a large value of q we reproduce in the pure case several known exact results the structure of the phase diagrams for all the values of q is obtained with a good precision the massless spin wave like phase which evolves into the kosterlitz touless phase for occurs around in agreement with the well known exact result the structure of the phase diagrams in the diluted case is qualitatively similar to that obtained from the pure model the massless spin wave like phase resists in an interval of concentration p which increases for the large values of q

a bellman approach for regional optimal control problems in mathbb r n this article is a continuation of a previous work where we studied infinite horizon control problems for which the dynamic running cost and control space may be different in two half spaces of some euclidian space mathbb r n in this article we extend our results in several directions i to more general domains ii to consideration of finite horizon control problems iii to weakening the controllability assumptions we use a bellman approach and our main results are to identify the right hamilton jacobi bellman equation and in particular the right conditions to be put on the interfaces separating the regions where the dynamic and running cost are different and to provide the maximal and minimal solutions as well as conditions for uniqueness we also provide stability results for such equations

proximal analysis and the minimal time function under general hypotheses on the target set s and the dynamics of the system we show that the minimal time function ts cdot is a proximal solution to the hamilton jacobi equation uniqueness results are obtained with two different kinds of boundary conditions a new propagation result is proven and as an application we give necessary and sufficient conditions for ts cdot to be lipschitz continuous near s a petrov type modulus condition is also shown to be sufficient for continuity of ts cdot near s

the critical behaviour of self dual z n spin systems finite size scaling and conformal invariance this paper is concerned with the critical properties of a family of self dual two dimensional z n models whose bulk free energy is exactly known at the self dual point the author s analysis is performed by studying the finite size behaviour of the corresponding one dimensional quantum hamiltonians which also possess an exact solution at their self dual point by exploring finite size scaling ideas and the conformal invariance of the critical infinite system the author calculates for n up to 8 the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra the results strongly suggest that the z n quantum field theory of zamolodchikov and fateev is the underlying field theory associated with these statistical mechanical systems the author also tests for the z 5 case the conjecture that these models correspond to the bifurcation points in the phase diagram of the general z n spin model where a massless phase originates

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

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

critical and off critical conformal analysis of the ising quantum chain in an imaginary field the critical curve with central charge c 22 5 of the ising quantum chain in an imaginary longitudinal field is calculated by finite size extrapolation of the yang lee square root branch point position various power laws are found the field dependence of the off criticality mass scale and the four lowest off criticality scaling functions are calculated the latter turn out to be universal along the critical line and agree very well with the results of the truncated hilbert space method for the c 22 5 continuum theory the author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic bethe ansatz result the scattering phaseshift calculated by luscher s method 1988 reproduces the cardy mussardo minimal solution 1989 for the s matrix if a boundary defect is introduced the only non trivial primary field decouples so that the spectrum consists only of the tower of the unit operator in the antiferromagnetic regime with a non staggered imaginary field the author finds boundary dependent lines where the gap vanishes

q state potts models in hamiltonian field theory for q 4 in 1 1 dimensions finite lattice sequence extrapolation methods are applied to the q state potts models in 1 1 d hamiltonian field theory thermal exponents nu 0 71 or 0 02 alpha 0 53 or 0 02 are obtained for the four state model methods are discussed for treating the first order transitions for q 4 and a sequence extrapolation is used to estimate the latent heat exact analytic formulae are derived for the ground state energy the mass gap and the latent heat at the transition point these are compared with the numerical results

running wave solutions of the two dimensional sine gordon equation after the lamb substitution the 2d sine gordon equation was solved three classes of solutions for this equation were found ad hoc the 2d sine gordon equation was reduced to an algebraic system consisting of three equations the solutions given of the 2d sine gordon equation are a generalization of the solutions of the 1d sine gordon equation they are also a generalization of the solutions of the equation phi xx yy sin phi x y

bulk and surface critical behaviour at the conformal invariant point in the z5 model the author determines the bulk thermal and magnetic exponents at the conformal invariant point in the z5 model with a monte carlo based finite size calculation the results are in good agreement with the predictions of zamolodchikov and fateev 1985 the author also argues that the n 1 surface magnetic exponents xh j in the zn model are given by j n j n j 1 n 1 this prediction is also verified in a monte carlo finite size calculation for the z5 model

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

finite size studies of the ashkin teller model the authors study the ashkin teller model in the time continuous hamiltonian version finite size scaling is used to calculate the magnetic gamma m electric gamma p and thermal alpha and nu critical exponents for several values of the coupling constant lambda the results confirm the believed extended scaling relations and suggest a conjecture relating the mass gap amplitudes and critical indices in the hamiltonian context

correlation functions of disorder fields and parafermionic currents in zn ising models we study the correlation functions of parafermionic currents and disorder fields in the zn symmetric conformal field theory perturbed by the first thermal operator following the ideas of al zamolodchikov zamolodchikov 1991 nucl phys b 348 619 zamolodchikov 1990 nucl phys b 342 695 and zamolodchikov 1995 int j mod phys a 10 1125 we develop for the correlation functions the conformal perturbation theory at small scales and the form factors spectral decomposition at large ones for all n there is an agreement between the data at the intermediate distances we consider the problems arising in the description of the space of scaling fields in perturbed models such as null vector relations equations of motion and a consistent treatment of fields related by a resonance condition

discrete mass spectrum of z n spin systems perturbed by a thermal field the author calculates numerically the lowest masses in the discrete mass spectrum of the z n fateev zamolodchikov spin model 1985 perturbed by a thermal operator the mass ratios obtained are in agreement with those derived from the s matrix introduced by koberle and swieca 1979

lyapunov functions on product spaces and stability theory of delay differential equations if we examine the lyapunov functionals constructed for all the examples that have been discussed so far in the literature we find that the investigators inadvertently employ a combination of a lyapunov function and a functional in such a way that the corresponding derivative can be estimated suitably without demanding minimal classes of functions or the knowledge of solutions as in the case of lyapunov functions or lyapunov functionals respectively 3 61 the method of lyapunov functionals however demands the knowledge of solutions of delay differential equations considered and consequently the discussion of examples is not really in the spirit of the method this observation leads us to develop the method of lyapunov functions on product spaces for studying stability properties of equations with delay where except conceptually the knowledge of solutions is not demanded we shall also develop stability theory in terms of two measures which unifies several known stability concepts our presentation demonstrates the advantage of utilizing lyapunov functions on product spaces 391 0022 247x 91 3 00