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

corrections to finite size scaling for quantum chains the authors compute the leading order correction to the energy gap at the critical point for the anisotropic ising and the z3 potts quantum chains periodic anti periodic and free boundary conditions are considered for the ising case the corrections are independent of the boundary conditions in the z3 case the correction is the same for anti periodic and free boundary conditions but is 2 5 smaller for the periodic case the authors have no explanation for this phenomenon

the relation between amplitudes and critical exponents in finite size scaling the relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two dimensional systems as a function of finite size is investigated and generalised in two ways correlations more general than those of order order type are included a form appropriate for anisotropic systems is proposed the authors present a exact results for the ising and gaussian models and b numerical results for the symmetric eight vertex baxter continuous q state potts and continuous n component cubic models

universal finite size scaling amplitudes on a torus for the triangular ising lattice gas recently the author suggested studying the scaling properties of complex two dimensional phase transitions by calculating the universal amplitudes of interfacial free energies on a torus geometry employing monte carlo simulations he tests this method at the 3 3 phase transition in the ising lattice gas on a triangular lattice belonging to the chiral three state potts universality class the interfacial free energies for this lattice gas model can be written in terms of free energy differences between systems with different sizes the algorithm to calculate these free energy differences is presented using lattice sizes up to 18 18 and modest computing effort he finds that the universal amplitude a 1 220 02 in accordance with the exact analytical value for the non chiral three state potts model a 1 203291 which he determined earlier

convexity properties of harmonic measures it is shown that any convex combination of harmonic measures where u1 uk are relatively compact open neighborhoods of a given point d2 can be approximated by a sequence of harmonic measures such that each wn is an open neighborhood of x in u1uk this answers a question raised in connection with jensen measures moreover it implies that for every green domain x containing x the extremal representing measures for x with respect to the convex cone of potentials on x these measures are obtained by balayage of the dirac measure at x on borel subsets of x are dense in the compact convex set of all representing measures this is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner these results which are presented simultaneously for the classical potential theory and for the theory of riesz potentials can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property the methods developed for this purpose continuous balayage on increasing families of compact sets approximation using scattered sets with small capacity finally lead to answers even in a very general potential theoretic setting covering a wide class of second order partial differential operators uniformly elliptic or in divergence form or sums of squares of vector fields satisfying hormander s condition for example sub laplacians on stratified lie algebras

stopping distributions for right processes letx be a transient right process for which semipolar sets are polar we characterize the measures which can arise as the distribution ofx t witht a non randomized stopping time

correlation length in ising strips with free and fixed boundary conditions the correlation length of boundary spins in the ising model defined on strips of triangular lattice with free boundary conditions is determined with an efficient numerical procedure based on the star triangle transformation in the case of isotropic critical interactions the extrapolated amplitude of the correlation length is in excellent agreement with the value 2 pi eta sub predicted by conformal invariance an analytical formula for the amplitude in strips with anisotropic interactions is proposed fixing the spins on one edge reduces the amplitude of the correlation length on the other edge by a factor 1 2 the convergence of phenomenological renormalisation with free boundary conditions is studied

finite size scaling in strips antiperiodic boundary conditions the exact values of the universal amplitude a relating the correlation length xi approximately l a to the strip width l at the critical point of ising and 3 state potts models are obtained for the case of antiperiodic or twisted boundary conditions predictions are also made for the finite size scaling behaviour of the interfacial tension

finite size scaling for directed bond percolation with and without cycles on a triangular lattice the authors apply finite size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic directed cyclic and undirected bond percolation on a triangular lattice the results are in good agreement with known estimates and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases as well as the distinction between locally directed and fully isotropic problems in addition the authors draw tentative comments on the extension to directed problems of the relation a pi eta where a is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point known to hold for various isotropic systems

observer based learning control for a class of nonlinear systems with time varying parametric uncertainties in this note a new learning control approach combined with state estimation is developed to perform output tracking problems where the state information is not available by virtue of the learning capability the control mechanism is able to handle a class of rapid time varying parametric uncertainties which are periodic and the only prior knowledge is the periodicity two classes of system nonlinearities are taken into account the first class is the global lipschitz continuous functions of the unknown state variables and the second class is the local lipschitz continuous functions of the accessible output variables to facilitate the learning control design and property analysis the lyapunov like energy function is employed which allows the incorporation of any available system knowledge henceforth the new learning control approach widens the application scope comparing with the repetitive type learning control

density of extremal measures in parabolic potential theory it is shown that for the heat equation on mathbb r d times mathbb r d 1 any convex combination of harmonic caloric measures mu x u 1 dots mu x u k where u 1 u k are relatively compact open neighborhoods of a given point x can be approximated by a sequence mu x w n n in mathbb n of harmonic measures such that each w n is an open neighborhood of x in u 1 cup dots cup u k moreover it is proven that for every open set u in mathbb r d 1 containing x the extremal representing measures for x with respect to the convex cone of potentials on u these measures are obtained by balayage with respect to u of the dirac measure at x on borel subsets of u are dense in the compact convex set of all representing measures since essential ingredients for a proof of corresponding results in the classical case or more general elliptic situations see hansen and netuka in adv math 218 4 1181 1223 2008 are not available for the heat equation an approach heavily relying on the transit character of the hyperplanes mathbb r d times c c in mathbb r is developed in fact the new method is suitable to obtain convexity results for limits of harmonic measures and the density of extremal representing measures on x x times mathbb r for practically every space time structure which is given by a sub markov semigroup p t t 0 on a space x such that there are strictly positive continuous densities t x y mapsto p t x y with respect to a non atomic measure on x in particular this includes many diffusions and corresponding symmetric processes given by heat kernels on manifolds and fractals moreover the results may be applied to restrictions of the space time structure on arbitrary open subsets

universal functions in finite size scaling a numerical study in the ising and potts q 3 universality class the universality of the correlation length order parameter and susceptibility amplitudes is studied in the vicinity of the critical point of two dimensional models on strips with periodic boundary conditions for the ising and potts q 3 universality class the results confirm privman fisher conjecture 1984

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

on the existence of evans potentials it is shown that for every noncompact parabolic riemannian manifold x and every nonpolar compact k in x there exists a positive harmonic function on x setminus k which tends to infty at infinity this is trivial for mathbb r easy for mathbb r 2 and known for parabolic riemann surfaces in fact the statement is proven more generally for any noncompact connected brelot harmonic space x where constants are the only positive superharmonic functions and for every nonpolar compact set k there is a symmetric positive green function for x setminus k this includes the case of parabolic riemannian manifolds without symmetry however the statement may fail this is shown by an example where the underlying space is a graph the union of the parallel half lines left 0 infty right times 0 left 0 infty right times 1 and the line segments n times 0 1 n 0 1 2 dots

finite size scaling and the two dimensional xy model shows that the finite size scaling assumption and particularly a conjectured relation involving the exponent eta are valid in the low temperature phase of the 2d xy model order by order in a perturbative expansion of the two point correlation function

an iterative learning control for a class of partially feedback linearizable systems a class of single input single output nonlinear systems which are partially linearizable by state feedback is considered feedback linearizable systems are included in such a class no parametrization is required for the uncertainties which are required to satisfy the matching condition periodic reference signals with known period t are to be tracked by the output provided that known bounding functions on the uncertainties are available a state feedback iterative learning control is designed which achieves asymptotic output tracking and guarantees bounded closed loop signals from any intial condition the novel control tecnhique is illustrated for a single link robot arm

repetitive learning control for a class of partially linearizable uncertain nonlinear systems the repetitive learning control for a class of partially feedback linearizable systems is considered the signal to be tracked is a general periodic function with known period the considered system especially the unlinearizable subsystem of it contains unknown functions only the desired state variables of the linearizable subsystem can be used in control law design first some system constants related to the desired system and error system are presented then based on these system constants a state feedback repetitive control algorithm is proposed which guarantees that all the signals in the closed loop systems are bounded and the tracking error converges to zero asymptotically a simulation example is presented to show the utility of the proposed control algorithm

on repetitive learning control for periodic tracking tasks in this note a repetitive learning control rlc approach is proposed to deal with periodic tracking tasks for nonlinear dynamical systems with nonparametric uncertainties we address two fundamental issues associated with the learning control methodology the existence of the solution and learning convergence property applying the existence theorem of the neutral differential difference equation and using lyapunov krasovskii functional the existence of the solution and learning convergence can be proven rigorously a further extension of the rlc to cascade systems is also explored

corrections to scaling and self duality in the restricted solid on solid model the corrections to scaling at and above the roughening transition tr in the restricted solid on solid model are studied by means of finite size scaling for semi infinite strips of width 2 to 10 it is shown that at a temperature tsd not far above the roughening temperature where the restricted solid on solid model is invariant under the duality transformation of the five state clock model the fugacity of the sine gordon operator must vanish for the dual truncated planar model this means that the core energy of the vortices diverges between ts and tr the corrections to scaling do not reach their asymptotic behaviour but are dominated by the contributions from the delta phi 4 operator associated with the restriction to steps of height one

harmonic measures for a point may form a square abstract let x be a green domain in r d d 2 x x and let m x p x denote the compact convex set of all representing measures for x recently it has been proven that the set of harmonic measures x u u open in x x u which is contained in the set of extreme points of m x p x is dense in m x p x in this paper it is shown that m x p x is not a simplex and hence not a poulsen simplex this is achieved by constructing open neighborhoods u 0 u 1 u 2 u 3 of x such that the harmonic measures x u 0 x u 3 are pairwise different and x u 0 x u 2 x u 1 x u 3 in fact these measures form a square with respect to a natural l 2 structure since the construction is mainly based on having certain symmetries it can be carried out just as well for riesz potentials the heisenberg group or any stratified lie algebra and the heat equation or more general parabolic situations

can one hear the dimension of a fractal we consider the spectrum of the laplacian in a bounded open domain of n with a rough boundary i e with possibly non integer dimension and we discuss a conjecture by m v berry generalizing weyl s conjecture then using ideas mark kac developed in his famous study of the drum we give upper and lower bounds for the second term of the expansion of the partition function the main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on minkowski dimensions and on minkowski measures rather than on haussdorff ones

free energy amplitude in finite size scaling the baxter model the critical amplitude a0 of the free energy density of the baxter model is studied numerically on strips with widths l 2 12 as a function of the four spin interaction k4 it remains almost constant near to double the exactly known ising result a0 2a0ising pi 6 in the range of k4 values for which an extrapolation to infinite width is possible

stopped distributions for markov processes in duality let x and hat x be standard markov processes in duality on a state space e and assume that semipolar sets are polar let be a measure on e whose x measure potential u is finite we characterize the measures v on e which arise as the p distribution of x t for some non randomized stopping time t we then apply this result to characterize the measures v on e which satisfy v u u

directed percolation a finite size renormalisation group approach the finite size renormalisation group technique introduced by nightingale 1976 is applied to the directed percolation problem the decay of correlations is anisotropic in this model and finite size scaling is extended to treat such anisotropy precise estimates for critical exponents and percolation probabilities are obtained for site bond and site bond percolation on the square lattice with bonds directed along the positive axes both free boundary conditions for which the results converge linearly with 1 n as n to infinity and helical boundary conditions for which unexpectedly the results converge linearly with 1 n3 are considered

brief paper asymptotic rejection of asymmetric periodic disturbances in output feedback nonlinear systems this paper deals with asymptotic rejection of periodic disturbances which may have asymmetric basic wave patterns this class of disturbances covers asymmetric wave forms in the half period such as alternating sawtooth wave form some disturbances which are generated from nonlinear oscillation such as van de pol oscillators as well as disturbances with symmetric half period wave forms such as sinusoidal disturbances and triangular disturbances etc the systems considered in this paper can be transformed to the nonlinear output feedback form the amplitude and phase of the disturbances are unknown the novel concept of integral phase shift is introduced together with the newly introduced half period integration operator to investigate the invariant properties of asymmetric periodic disturbances they are used for the estimation of unknown disturbances in the systems together with observer design techniques to deal with nonlinearity the proposed control design with the disturbance estimation asymptotically rejects the unknown disturbance and ensures the overall stability of the system

spin wave expansion for the o 2 heisenberg spin model in d 1 dimensions spin wave expansions are carried out for the hamiltonian version of the o 2 heisenberg spin model and evaluated for the cases of the 1 1 dimensional linear chain and the 2 1 dimensional square and triangular lattices expressions are obtained for the bulk limits and finite size corrections to the ground state energy the one boson excitation energy the magnetization and susceptibility at low temperatures results are derived for the critical parameters in the 1 1 dimensional model and the goldstone mode parameters in the 2 1 dimensional models comparison is made with the finite size scaling predictions of fisher and privman 1985

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

on the symmetry of universal finite size scaling functions in anisotropic systems we postulate a symmetry of universal finite size scaling functions under a certain anisotropic scale transformation which connects the properties of a finite two dimensional 2d system at criticality with generalized aspect ratio 1 to a system with 1 the symmetry is formulated within a finite size scaling theory and expressions for several universal amplitude ratios are derived the predictions are confirmed within the exactly solvable weakly anisotropic 2d ising model and are checked within the strongly anisotropic 2d dipolar in plane ising model using monte carlo simulations

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

universal amplitudes in finite size scaling generalisation to arbitrary dimensionality the relationship between the correlation length and critical exponents in finite width strips in two dimensions is generalised to cylindrical geometries of arbitrary dimensionality d for d not 2 these correspond however to curved spaces the result is verified for the spherical model

comments on repetitive learning control for a class of partially linearizable uncertain nonlinear systems automatica 85 2017 397 404 abstract this correspondence points out connections between the results achieved in verrelli 2016 and the ones obtained in chen and liu 2017 problem formulation control structure and stability proof in chen and liu 2017 can be revisited and recast to reveal similarities that provide a new interpretation of such a paper in the light of verrelli 2016

a new periodic adaptive control approach for time varying parameters with known periodicity a new adaptive control approach is proposed to deal with nonlinear systems with a class of time varying parametric uncertainties which are in an unknown compact set periodic nonvanishing rapid time varying and the only prior knowledge is the periodicity unlike most continuous time adaptation laws which are of differential types in this work a periodic type adaptation law is introduced for continuous time systems the new adaptive controller updates the parameters and the control signal periodically in a pointwise manner over one entire period in the sequel achieves the asymptotic tracking convergence

a global robust iterative learning position control for current fed permanent magnet step motors the tracking control problem via state feedback for uncertain current fed permanent magnet step motors with non sinusoidal flux distribution and uncertain position dependent load torque is addressed a periodic reference signal of known period for the rotor position is required to be tracked a robust iterative learning control algorithm is designed which for any motor initial condition and without requiring any resetting procedure guarantees despite system uncertainties exponential convergence of the rotor position tracking error to a residual ball centered at the origin whose radius can be made arbitrarily small by properly setting the learning gain asymptotic convergence of the rotor position tracking error to zero a sufficient condition for the asymptotic estimation of the uncertain reference input achieving for compatible initial conditions perfect tracking is derived robustness with respect to a finite memory implementation of the control algorithm based on the piecewise linear approximation theory is shown to be guaranteed satisfactory performances of a discrete time implementation of the control algorithm are obtained in realistic simulations for the full order voltage fed motor

partially directed site percolation on the square and triangular lattices using the phenomenological renormalisation pr method the authors obtain the critical probability for partially directed site percolation on the square and triangular lattices pc 0 6317 or 0 006 and pc 0 5468 or 0 005 respectively they discuss the corrections to scaling in the calculation of the anisotropy exponent theta nu sub nu perpendicular to and the critical probability and show that it is necessary to work with at least two correction terms this is in contrast to the usual analysis where only one correction exponent is used in pr calculations

asymptotic state tracking in a class of nonlinear systems via learning based inversion describes a learning approach to asymptotic state tracking in a class of nonlinear systems the tracking problem considered concerns the case when the tracking error dynamics are described by a set of time varying nonlinear differential equations which are periodic in time with a known period our iterative update scheme is based on the specific property that the learning system tends to oscillate in steady state in fact our approach extends in a very natural manner the idea of the well known iterative learning control for the case of finite time tracking problems to the case of infinite time asymptotic tracking problems the best advantage of the proposed leaning approach is that it is computationally simple and does not require one to solve any complicated equations based on full system dynamics we explore the conditions under which a periodic nonlinear system exhibits a steady state oscillation our work also can be viewed to provide a learning based solution to the input state inversion problem the generality and practicality of our work is demonstrated through rigorous performance analysis and simulation using a robot manipulator

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

adaptive asymptotic rejection of unmatched general periodic disturbances in output feedback nonlinear systems this note deals with asymptotic rejection of periodic disturbances in a class of nonlinear systems via output feedback the disturbance undertaken is unmatched but the period of the disturbance is known a novel method for estimating periodic disturbances is proposed which ensures convergence of the estimation error to zero by applying such estimation a controller is constructed such that all the signals in the closed loop are bounded and the output converges to zero a numerical example is presented to show the utility of the results presented in the note

radial fredholm perturbation in the two dimensional ising model and gap exponent relation we consider concentric circular defects in the two dimensional ising model which are distributed according to a generalized fredholm sequence i e at exponentially increasing radii this type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation the critical exponent of the local magnetization is obtained through finite size scaling using a corner transfer matrix approach in the extreme anisotropic limit it varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial hilhorst van leeuwen model through a conformal mapping of the system onto a strip the gap exponent relation is shown to remain valid for such an aperiodic defect

random tilings with quasicrystal order transfer matrix approach the random tiling of the plane by a set of objects e g rhombi related by rotational e g tenfold symmetries is a paradigm for the formation of quasiperiodic order due to entropy such tilings are mapped to a higher dimensional space where they form hypersurfaces analogous to the interfaces in a solid on solid model the author argues that the fluctuations of the hypersurface should be described by a gradient squared free energy of entropic origin this implies quasi long range order in d 2 he shows how the random tiling can be decomposed into layers defines a transfer matrix and give prescriptions for using this method to determine numerically the stiffness of the gradient free energy

adaptive control of nonlinearly parameterized systems the smooth feedback case studies global adaptive control of nonlinearly parameterized systems with uncontrollable linearization using a parameter separation technique and the tool of adding a power integrator we develop a feedback domination design approach for the explicit construction of a smooth adaptive controller that solves the problem of global state regulation in contrast to the existing results in the literature a key feature of our adaptive regulator is its minimum order property namely no matter how big the number of unknown parameters is the order of the dynamic compensator is identical to one and is therefore minimal as a consequence global state regulation of feedback linearizable systems with nonlinear parameterization is achieved by one dimensional adaptive controllers without imposing any extra e g convex concave conditions on the unknown parameters

phenomenological renormalisation of the self avoiding walk in two dimensions phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self avoiding walk problem on a square lattice a transfer matrix technique is developed for the polymer problem the results indicate that flory s value v 0 75 is true in two dimensions to extremely high accuracy

basis vector importance sampling for hamiltonian lattice spectrum calculations different numerical methods for accurate calculation of low lying eigenvalues of lattice hamiltonians are proposed and critically compared a dynamical procedure called basis vector importance sampling is shown to select the relevant subspace of the hilbert space very effectively this method is used to compute the mass gap of o 2 symmetric quantum chains up to a length of nine sites kosterlitz thouless type freezing transitions of z p symmetric chains are also studied via the spectrum of quantum kinks

hunting for the central charge of the virasoro algebra six and eight state spin models in two dimensions conformal invariance has important implications for the finite size scaling properties of the spectra of transfer matrices and quantum chains at the critical point some relations between the finite size scaling amplitudes are obtained which can be used as a test of conformal invariance and implicitly to distinguished between second order and first order phase transitions the numerical values of the scaling amplitudes give the scale dimensions of various operators six and eight state self dual quantum chains with cubic symmetries are considered at the critical point for three values of the coupling constant the systems are found to be conformal invariant and estimates for several critical exponents are obtained based on the approximate values of the critical exponent one tries to find the values of the corresponding central charges of the virasoro algebras

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

on the validity of the gap exponent relation in ising models with many defect lines numerical evidence is presented to support the conjecture that the gap exponent relation remains valid for multi defect ising systems where exact solutions are unavailable predictions for exponents corresponding to arbitrary lattice anisotropy are given

critical behaviour and logarithmic corrections of a quantum model with three spin interaction extended finite size scaling and series expansion methods are used to determine the critical properties of the one dimensional quantum ising model with three spin interaction by supposing the presence of logarithmic corrections the universality class of the model is investigated the results obtained by different methods support the conjecture that the model belongs to the same universality class as the four state potts model

finite size scaling of the free energy levels in the ising and potts q 3 universality the finite size scaling of the free energy levels of privman and fisher 1984 is tested numerically on two dimensional lattice strips in the ising and q 3 potts universality the universal amplitudes and the free energy per site are obtained

adaptive repetitive control for a class of nonlinearly parametrized systems in this note lyapunov based adaptive repetitive control is presented for a class of nonlinearly parametrized systems through the use of an integral lyapunov function the controller singularity problem is elegantly solved as it avoids the nonlinear parametrization from entering into the adaptive control and repetitive control global stability of the adaptive system and asymptotic convergence of the tracking error are established and tracking error bounds are provided to quantify the control performance both partially and fully saturated learning laws are analyzed in detail and compared analytically

the 1 1 do 2 model a finite lattice analysis finite size methods and the theory of conformal invariance are applied to the 1 1 do 2 or x y planar model the critical point is found to be xc approximately 2 0 the value sigma 0 501 or 0 005 is obtained for the index governing the critical divergence of the correlation length and the conformal anomaly is confirmed to be c 1 the kosterlitz thouless features of a low temperature phase with zero spontaneous magnetisation but algebraic decay of correlations together with a standard high temperature phase are observed the critical exponent eta is determined as a function of coupling constant and compared with theoretical expectations

asymptotic rejection of general periodic disturbances in output feedback nonlinear systems this note deals with asymptotic rejection of general periodic disturbances in nonlinear systems a novel approach based on integrations over the half of disturbance period is proposed to estimate disturbances which may not be harmonic or even smooth the proposed algorithm uses the period and the knowledge of the basic disturbance patterns while the amplitude and phase of the disturbances are unknown and they are estimated based on the estimated disturbance the disturbance rejection algorithm is then proposed for linear systems and nonlinear systems in the output feedback format which asymptotically rejects the periodical disturbances the proposed algorithm extends asymptotic rejection of unknown disturbances from sinusoidal disturbances to a more general class of periodical disturbances which includes zero mean square waves symmetric triangular waves etc

finite size scaling of the quantum ising chain with periodic free and antiperiodic boundary conditions the authors give exact results for the energy spectrum of a chain of n ising spins in a transverse field with periodic free and antiperiodic boundary conditions the dependence of the energy gaps on boundary conditions is compatible with predictions of conformal invariance for correlation lengths in two dimensional strips the feasibility of calculating surface critical indices using phenomenological renormalisation with free boundary conditions and the convergence for large n are discussed