an sos based control lyapunov function design for polynomial fuzzy control of nonlinear systems this paper deals with a sum of squares sos based control lyapunov function clf design for polynomial fuzzy control of nonlinear systems the design starts with exactly replacing smooth nonlinear systems dynamics with polynomial fuzzy models which are known as universal approximators next global stabilization conditions represented in terms of sos are provided in the framework of the clf design i e a stabilizing controller with nonparallel distributed compensation form is explicitly designed by applying sontag s control law once a clf for a given nonlinear system is constructed furthermore semiglobal stabilization conditions on operation domains are derived in the same fashion as in the global stabilization conditions both global and semiglobal stabilization problems are formulated as sos optimization problems which reduce to numerical feasibility problems five design examples are given to show the effectiveness of our proposed approach over the existing linear matrix inequality and sos approaches

self averaging in the statistical mechanics of some lattice models we discuss self averaging of thermodynamic properties in some random lattice models in particular we investigate when self averaging in an almost sure sense of the free energy implies self averaging of the energy and heat capacity and we discuss the connection between self averaging in the almost sure sense and self averaging in an l p sense under quite general conditions we show that the average of the finite size heat capacity converges to the second derivative of the limiting quenched average free energy we consider the application of these ideas to the problem of adsorption of a random copolymer at a surface and to some related systems

new construction of solvable lattice models including an ising model in a field in this letter we report a new construction to obtain restricted solid on solid rsos models out of loop models the method is a generalization of ideas developed by owczarek and baxter and by pasquier in particular we consider a solvable o n model and point out that some of the rsos models thus obtained admit an off critical extension among these models we find a spin 1 ising model which is solvable not only at the critical point but also in a fieldlike deviation away from it we calculate the critical exponent ensuremath delta 15 directly from the relation between the free energy and the field this is the first determination of this exponent without the use of scaling relations

conjugate lyapunov functions for saturated linear systems based on a recent duality theory for linear differential inclusions ldis the condition for stability of an ldi in terms of one lyapunov function can be easily derived from that in terms of its conjugate function this paper uses a particular pair of conjugate functions the convex hull of quadratics and the maximum of quadratics for the purpose of estimating the domain of attraction for systems with saturation nonlinearities to this end the nonlinear system is locally transformed into a parametertized ldi system with an effective approach which enables optimization on the parameter of the ldi along with the optimization of the lyapunov functions the optimization problems are derived for both the convex hull and the max functions and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids a numerical example demonstrates the effectiveness of this paper s methods

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

composite quadratic lyapunov functions for constrained control systems a lyapunov function based on a set of quadratic functions is introduced in this paper we call this lyapunov function a composite quadratic function some important properties of this lyapunov function are revealed we show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids these results are used to study the set invariance properties of continuous time linear systems with input and state constraints we show that for a system under a given saturated linear feedback the convex hull of a set of invariant ellipsoids is also invariant if each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators then their convex hull can also be made invariant by the same actuators for a set of ellipsoids each invariant under a separate saturated linear feedback we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant

critical behavior of a three dimensional random bond ising model using finite time scaling with extensive monte carlo renormalization group method we have investigated the critical behavior of a three dimensional random bond ising model for a series of the disorder strength by a finite time scaling combining with monte carlo renormalization group method in the presence of a linearly varying temperature the method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures the static exponents obtained agree well with most existing results verify both the hyperscaling and the rushbrooke scaling laws and their combined scaling law which in turn validate their asymptotic nature and corroborate the universality of the relevant random fixed point with respect to the forms of disorder the dynamic critical exponent z is estimated to 2 114 51 which is compatible with those obtained from experiments and renormalization group analyses the exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point they also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover our results demonstrate the effectiveness of the finite time scaling method

percolation thresholds and fractal dimensions for square and cubic lattices with long range correlated defects we study long range power law correlated disorder on square and cubic lattices in particular we present high precision results for the percolation thresholds and the fractal dimension of the largest clusters as a function of the correlation strength the correlations are generated using a discrete version of the fourier filtering method we consider two different metrics to set the length scales over which the correlations decay showing that the percolation thresholds are highly sensitive to such system details by contrast we verify that the fractal dimension d f is a universal quantity and unaffected by the choice of metric we also show that for weak correlations its value coincides with that for the uncorrelated system in two dimensions we observe a clear increase of the fractal dimension with increasing correlation strength approaching d f 2 the onset of this change does not seem to be determined by the extended harris criterion

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

doubly stochastic matrices whose powers eventually stop abstract in this note we characterize doubly stochastic matrices a whose powers a a 2 a 3 eventually stop i e a p a p 1 for some positive integer p the characterization enables us to determine the set of all such matrices

magnetic critical behavior of two dimensional random bond potts ferromagnets in confined geometries we present a numerical study of two dimensional random bond potts ferromagnets the model is studied both below and above the critical value qc 4 which discriminates between second and first order transitions in the pure system two geometries are considered namely cylinders and square shaped systems and the critical behavior is investigated through conformal invariance techniques that were recently shown to be valid even in the randomness induced second order phase transition regime q 4 in the cylinder geometry connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes that is characteristic of the disordered fixed point the scaling dimensions then follow from the exponential decay of correlations along the strip monte carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice but the data are then perturbed by strong surface effects the conformal mapping of a semi infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions the techniques are applied to different values of q in the range 3 64

weak randomness for large q state potts models in two dimensions we have studied the effect of weak randomness on q state potts models for q 4 by measuring the central charges of these models using transfer matrix methods we obtain a set of new values for the central charges and then show that some of these values are related to one another by a factorization law

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

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

on the universality class of the 3d ising model with long range correlated disorder abstract we analyze a controversial topic about the universality class of the three dimensional ising model with long range correlated disorder whereas both theoretical and numerical studies agree on the validity of the extended harris criterion a weinrib b i halperin phys rev b 27 1983 413 and indicate the existence of a new universality class numerical values of the critical exponents found so far differ considerably to resolve this discrepancy we perform extensive monte carlo simulations of a 3d ising model with non magnetic impurities being arranged in a form of lines along randomly chosen axes of a lattice the swendsen wang algorithm is used alongside with a histogram reweighting technique and finite size scaling analysis to evaluate the values of critical exponents governing magnetic phase transition our estimates for these exponents differ from both previous numerical simulations and are in favor of a non trivial dependency of the critical exponents on the peculiarities of long range correlation decay

properties of the composite quadratic lyapunov functions a composite quadratic lyapunov function introduced recently was shown to be very useful in the study of set invariance properties for linear systems with input and state constraints and for systems with a class of convex concave nonlinearities in this note more properties about this function are revealed in particular we study the continuity of the optimal parameter involved in this function this continuity is crucial in the construction of a continuous feedback law which makes the convex hull of a group of ellipsoids invariant

stability and performance for saturated systems via quadratic and nonquadratic lyapunov functions in this paper we develop a systematic lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration the only assumptions we make about the system are well posedness of the algebraic loop and local stability problems to be considered include the estimation of the domain of attraction the reachable set under a class of bounded energy disturbances and the nonlinear l2 gain the regional analysis is established through an effective treatment of the algebraic loop and the saturation deadzone function this treatment yields two forms of differential inclusions a polytopic differential inclusion pdi and a norm bounded differential inclusion ndi that contain the original system adjustable parameters are incorporated into the differential inclusions to reflect the regional property the main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain lyapunov function where the pdi or the ndi is valid with quadratic lyapunov functions conditions for stability and performances are derived as linear matrix inequalities lmis to obtain less conservative conditions we use a pair of conjugate non quadratic lyapunov functions the convex hull quadratic function and the max quadratic function these functions yield bilinear matrix inequalities bmis as conditions for stability and guaranteed performance level the bmi conditions cover the corresponding lmi conditions as special cases hence the bmi results are guaranteed to be as good as the lmi results in most examples the bmi results are significantly better than the lmi results

monte carlo study of the triangular blume capel model under bond randomness the effects of bond randomness on the universality aspects of a two dimensional d 2 blume capel model embedded in the triangular lattice are discussed the system is studied numerically in both its first and second order phase transition regimes by a comprehensive finite size scaling analysis for a particularly suitable value of the disorder strength we find that our data for the second order phase transition emerging under random bonds from the second order regime of the pure model are compatible with the universality class of the two dimensional 2d random ising model furthermore we find evidence that the second order transition emerging under bond randomness from the first order regime of the pure model belongs again to the same universality class although the first finding reinforces the scenario of strong universality in the 2d ising model with quenched disorder the second is in difference from the critical behavior emerging under randomness in the cases of the ex first order transitions of the potts model finally our results verify previous renormalization group calculations on the blume capel model with disorder in the crystal field coupling

on the optimal parameter of the composite laplacian quadratics function abstract recently the composite quadratic lyapunov function has been extended to study of multi agent systems leading to the so called composite laplacian quadratics clqs function compared with quadratic lyapunov functions the clqs function can yield a larger convergence region and is particularly useful in stabilization of multi agent systems with complex dynamics such as differential inclusions in the definition of the clqs function an optimal vector parameter plays a critical role in determining the value of the clqs function and in constructing stabilization laws derived from the clqs function this paper focuses on the properties of the optimal parameter of the clqs function the uniqueness of the optimal parameter is established a distributed computation approach is further proposed which is useful in computing the optimal parameter the robustness issue of the optimal parameter is also investigated for a multi agent system described by linear differential inclusions finally a numerical example is provided to validate the proposed theoretical results

brief paper non conservative matrix inequality conditions for stability stabilizability of linear differential inclusions this paper shows that the matrix inequality conditions for stability stabilizability of linear differential inclusions derived from two classes of composite quadratic functions are not conservative it is established that the existing stability stabilizability conditions by means of polyhedral functions and based on matrix equalities are equivalent to the matrix inequality conditions this implies that the composite quadratic functions are universal for robust possibly constrained stabilization problems of linear differential inclusions in particular a linear differential inclusion is stable stabilizable with without constraints iff it admits a lyapunov control lyapunov function in these classes examples demonstrate that the polyhedral functions can be much more complex than the composite quadratic functions to confirm the stability stabilizability of the same system

probability distribution of persistent spins in an ising chain we study the probability distribution q n t of n t the fraction of spins unflipped up to time t in an ising chain with ferromagnetic interactions the distribution shows a peak at n nmax and in general is non gaussian and asymmetric in nature however for n nmax it shows a gaussian decay data collapse can be obtained when q n t l versus n nmax l is plotted with 0 45 and 0 6 interestingly nmax t shows different behaviour compared to n t p t the persistence probability which follows the well known behaviour p t t a quantitative estimate of the asymmetry and non gaussian nature of q n t is made by calculating its skewness and kurtosis

lattice realizations of unitary minimal modular invariant partition functions the conformal spectra of the critical dilute a d e lattice models are studied numerically the results strongly indicate that in branches one and two these models provide realizations of the complete a d e classification of unitary minimal modular invariant partition functions given by cappelli itzykson and zuber 1987 in branches three and four the results indicate that the modular invariant partition functions factorize similar factorization results are also obtained for two colour lattice models

brief paper nonlinear control design for linear differential inclusions via convex hull of quadratics this paper presents a nonlinear control design method for robust stabilization and robust performance of linear differential inclusions ldis a recently introduced non quadratic lyapunov function the convex hull of quadratics will be used for the construction of nonlinear state feedback laws design objectives include stabilization with maximal convergence rate disturbance rejection with minimal reachable set and least l 2 gain conditions for stabilization and performances are derived in terms of bilinear matrix inequalities bmis which cover the existing linear matrix inequality lmi conditions as special cases numerical examples demonstrate the advantages of using nonlinear feedback control over linear feedback control for ldis it is also observed through numerical computation that nonlinear control strategies help to reduce control effort substantially

a universal amplitude ratio for the q 4 potts model from a solvable lattice model the universal amplitude ratio r for the q 4 state potts model in two dimensions is determined by using results for the dilute a model in regime 1 the nature of the relationship between the potts model and the dilute a model is discussed

order parameters of the dilute a models the free energy and local height probabilities of the dilute a models with borken 2 symmetry are calculated analytically using inversion and corner transfer matrix methods these models possess four critical branches the first two branches provide new realizations of the unitary minimal series and the other two branches give a direct product of this series with an ising model we identify the integrable perturbations which move the dilute a models away from the critical limit generalized order parameters are defined and their critical exponents extracted the associated conformal weights are found to occur on the diagonal of the relevant kac table in an appropriate regime the dilute a3 model lies in the universality class of the ising model in a magnetic field in this case we obtain the magnetic exponent 15 directly without the use of scaling relations

susceptibility amplitudes for the three and four state potts models abstract the finite lattice method of series expansions is used to calculate a number of extended series for susceptibilities of the three and four state potts models on the square lattice we analyse all these series and estimate their critical amplitudes we resolve an uncertainty in amplitude ratios for high to low temperature susceptibilities and thus provide confirmation of field theoretic predictions

lyapunov function construction by linear programming in this paper the lyapunov function given by the infinity norm is considered a new proof of the theorem of molchanov and pyatintskii 1986 is given the presented proof enables us to give an explicit calculation of the matrix which was only demonstrated to exist in the original paper combining the obtained result with the idea of scaling faces of a polytope gives the method of fitting the shape of level surfaces of the lyapunov function given by the infinity norm to the specified demands

stabilization of switched systems via composite quadratic functions this paper investigates stabilization of switched systems by using a lyapunov function composed of a family of continuously differentiable functions switching laws are constructed by using the directional derivatives along the trajectories of the subsystems conditions for stabilization are established with careful consideration of sliding modes and directional derivatives along sliding modes three types of composite quadratic lyapunov functions the max of quadratics the min of quadratics and the convex hull of quadratics are used for deriving matrix conditions of stabilization and for constructing switching laws dual stabilization results are established with respect to the pair of conjugate lyapunov functions the max of quadratics and the convex hull of quadratics conditions of stabilization are derived as bilinear matrix inequalities and solved with lmi based numerical tools relationship between the newly derived conditions and some existing conditions are investigated it is observed that the min of quadratics which is nondifferentiable and nonconvex may be a more convenient tool than the other two types of functions which are convex and or differentiable numerical examples are used to demonstrate the synthesis results and the advantage of the composite quadratic functions over existing multiple lyapunov functions in particular better results have been obtained when the number of quadratic functions is greater than the number of subsystems

large q asymptotics of the random bond potts model we numerically examine the large q asymptotics of the q state random bond potts model special attention is paid to the parametrisation of the critical line which is determined by combining the loop representation of the transfer matrix with zamolodchikov s c theorem asymptotically the central charge seems to behave like c q 1 2 log 2 q o 1 very accurate values of the bulk magnetic exponent x 1 are then extracted by performing monte carlo simulations directly at the critical point as q infinity these seem to tend to a non trivial limit x 1 0 192 0 002

high temperature series analysis of the free energy and susceptibility of the 2d random bond ising model abstract we derive high temperature series expansions for the free energy and susceptibility of the two dimensional random bond ising model with a symmetric bimodal distribution of two positive coupling strengths j1 and j2 and study the influence of the quenched random bond disorder on the critical behavior of the model by analysing the series expansions over a wide range of coupling ratios j2 j1 covering the crossover from weak to strong disorder we obtain for the susceptibility with two different methods compelling evidence for a singularity of the form t 7 4 ln t 7 8 as predicted theoretically by shalaev shankar and ludwig for the specific heat our results are less convincing but still compatible with the theoretically predicted log log singularity

critical behavior of an even offspringed branching and annihilating random walk cellular automaton with spatial disorder a stochastic cellular automaton exhibiting a parity conserving class transition has been investigated in the presence of quenched spatial disorder by large scale simulations numerical evidence has been found that weak disorder causes irrelevant perturbation for the universal behavior of the transition and the absorbing phase of this model this opens up the possibility for experimental observation of the critical behavior of a nonequilibrium phase transition to absorbing state for very strong disorder the model breaks up into blocks with exponential size distribution and continuously changing critical exponents are observed for strong disorder the randomly distributed diffusion walls introduce another transition within the inactive phase of the model in which residual particles survive the extinction the critical dynamical behavior of this transition has been explored

piecewise lyapunov functions for robust stability of linear time varying systems in this paper we investigate the use of two term piecewise quadratic lyapunov functions for robust stability of linear time varying systems by using the so called s procedure and a special variable reduction method we provide numerically efficient conditions for the robust asymptotic stability of the linear time varying systems involving the convex combinations of two matrices an example is included to demonstrate the usefulness of our results

critical behavior of the pure and random bond two dimensional triangular ising ferromagnet we investigate the effects of quenched bond randomness on the critical properties of the two dimensional ferromagnetic ising model embedded in a triangular lattice the system is studied in both the pure and disordered versions by the same efficient two stage wang landau method in the first part of our study we present the finite size scaling behavior of the pure model for which we calculate the critical amplitude of the specific heat s logarithmic expansion for the disordered system the numerical data and the relevant detailed finite size scaling analysis along the lines of the two well known scenarios logarithmic corrections versus weak universality strongly support the field theoretically predicted scenario of logarithmic corrections a particular interest is paid to the sample to sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality

a numerical study of self averaging in adsorption of random copolymers and random surfaces numerical studies involving random copolymers and random surfaces assume self averaging of thermodynamic and metric properties of the systems to calculate different properties for the problem of adsorption of a random copolymer rigorous proofs regarding self averaging of some properties such as free energy in the thermodynamic limit n exist this says little about the extent of self averaging for finite size systems used in numerical studies for the problem of adsorption of a homopolymer on a random surface no analytical proofs regarding self averaging exist in this work assumptions of self averaging of thermodynamic and metric properties of a self avoiding walk model of random copolymer adsorption are tested via multiple markov chain monte carlo method numerical evidence is provided in support of self averaging of energy heat capacity and the z component of the self avoiding walk in different temperature intervals self averaging in energy of a homopolymer interacting with a random surface is also examined

quenched bond dilution in two dimensional potts models we report a numerical study of the bond diluted two dimensional potts model using transfer matrix calculations for different numbers of states per spin we show that the critical exponents at the random fixed point are the same as in self dual random bond cases in addition we determine the multifractal spectrum associated with the scaling dimensions of the moments of the spin spin correlation function in the cylinder geometry we show that the behaviour is fully compatible with the one observed in the random bond case confirming the general picture according to which a unique fixed point describes the critical properties of different classes of disorder dilution self dual binary random bond self dual continuous random bond

relaxed stabilization criterion for t s fuzzy systems by minimum type piecewise lyapunov function based switching fuzzy controller this paper proposes the minimum type piecewise lyapunov function based switching fuzzy controller that switches accompanying the piecewise lyapunov function by applying this switching fuzzy controller with the minimum type piecewise lyapunov function the relaxed stabilization criterion is obtained for continuous takagi sugeno t s fuzzy systems some conditions of the relaxed stabilization criterion are represented by bilinear matrix inequalities bmis which contain some bilinear terms as the product of a full matrix and a scalar according to the literature the path following method is very effective for this kind of bmi problem hence it is utilized to obtain solutions of the criterion xie et al in 1997 chose two types i e minimum type and maximum type of piecewise lyapunov functions as the lyapunov function candidates the reasons for why this study only chooses the minimum type piecewise lyapunov function as the lyapunov function candidate are illustrated moreover the numerical example shows the relaxation of the proposed criterion

a numerical technique for the stability analysis of linear switched systems in this paper the ray gridding approach a new numerical technique for the stability analysis of linear switched systems is presented it is based on uniform partitions of the state space in terms of ray directions which allow refinable families of polytopes of adjustable complexity to be examined for invariance in this framework the existence of a polyhedral lyapunov function that is common to a family of asymptotically stable subsystems can be checked efficiently via simple iterative algorithms the technique can be used to prove the stability of switched linear systems classes of linear time varying systems and linear differential lnclusions we also present preliminary results on another related problem namely the construction of multiple polyhedral lyapunov functions for the specification of stabilizing switching sequences for a switched system constructed from a family of stable linear subsystems

reach control problem for linear differential inclusion systems on simplices this technical note concerns the reach control problem for the dynamical systems represented by linear differential inclusions the goal is to derive conditions for the trajectories of a differential inclusion defined on a full dimensional simplex to reach exit facets in finite time using affine feedback to achieve this goal an invariance condition is firstly derived for lipschitz differential inclusions as an application of the obtained invariance condition the reach control problem rcp is solved for two classes of differential inclusions norm bounded linear differential inclusion and polytypic linear differential inclusion in the sense of strong and weak reachability respectively

finite size scaling study of two dimensional dilute potts models the author uses transfer matrices to calculate the free energy and various thermodynamic quantities of the two and three state potts models with random bond interactions at the critical temperature he verifies the harris criterion for the three state potts model and estimates the new random critical exponents for the ising case the results are consistent with randomness being irrelevant for the critical behaviour

two dimensional scaling in magnetic systems with non integer characteristic lengths we studied the ferromagnetic ising model on two dimensional finite systems with non integer characteristic lengths first we considered very long strips with a finite number i of complete rows and one partially filled row with probability x which is the two dimensional version of a layer by layer growth at fixed temperature when the characteristic length l i x increases the free energy per spin the specific heat and the magnetic susceptibility oscillate attaining relative extremes at integer l the oscillations in the free energy are interpreted as surface corrections related to the oscillations in the mean coordination number finite size scaling relations are not satisfied with continuous l but still hold for fixed x and variable i where the differences of mean widths are integers for fixed x the fits of the free energy give the conformal anomaly with very good accuracy we also studied strips with discretized gaussian distributions of widths with non integer means l and rms deviation in these structures the thermodynamic quantities vary monotonically with continuous l but some methods for calculating critical exponents do not work properly when generalized to continuous l we also obtain with good accuracy in these systems we discuss the possible implications of our results to real systems behaviour

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

some applications of doubly stochastic matrices abstract we discuss primarily two applications of doubly stochastic matrices and related matrices the first concerns a topic in communication theory called satellite switched time division multiple access systems and we attempt to illuminate some results which have occured in the engineering literature the second concerns a recent notion of a doubly stochastic automorphism of a graph we prove several new theorems about doubly stochastic automorphisms of certain classes of graphs and in particular obtain a theorem of which birkhoff s theorem for doubly stochastic matrices is a special case

self averaging distribution of pseudocritical temperatures and finite size scaling in critical disordered systems we evaluate by monte carlo simulations various singular thermodynamic quantities x for ensembles of quenched random ising and ashkin teller models the measurements are taken at t c and we study how the distributions p x and in particular their relative squared width r x over the ensemble depend on the system size l the ashkin teller model was studied in the regime where bond randomness is irrelevant and we found weak self averaging r x ensuremath sim l ensuremath alpha ensuremath nu ensuremath rightarrow 0 where ensuremath alpha l0 and ensuremath nu are the exponents of the pure model fixed point governing the transition for the site dilute ising model on a cubic lattice known to be governed by a random fixed point we find that r x tends to a constant as predicted by aharony and harris we tested whether this constant is universal however this constant is different for canonical and grand canonical disorder we identify the pseudocritical temperature of each sample i t c i l as the temperature at which the susceptibility reaches its maximal value ensuremath chi mathrm max the distribution of these sample dependent t c i l was investigated we found that its variance scales as ensuremath delta t c l 2 ensuremath sim l ensuremath 2 ensuremath nu our previously proposed finite size scaling ansatz for disordered systems was tested and found to hold we did observe deviations from a single function which imply that sample dependent scaling functions are needed these deviations are however relatively small and hence to obtain a fixed statistical error it may be more computationally efficient to measure ensuremath chi mathrm max than the commonly used ensuremath chi t c ensuremath infty

spin models on random lattices under certain conditions phase transitions in systems with quenched disorder are expected to exhibit a dierent behaviour than in the corresponding pure system here we discuss a series of monte carlo studies of a special type of such disordered systems namely spin models dened on quenched random lattices exhibiting geometrical disorder in the connectivity of the lattice sites in two dimensions we present results for the q state potts model on random tri valent 3 planar graphs which appear quite naturally in the dynamically triangulated random surface dtrs approach to quantum gravity as well as on poissonian random lattices of voronoi delaunay type both cases q64 and 4 are discussed which in the pure model without disorder give rise to second and rst order phase transitions respectively in three dimensions results for the ising model on poissonian random lattices are briey described we conclude with a comparison of the two types of connectivity disorder with the more standard case of bond disorder and a discussion of the distinguishing dierences c 2000 elsevier science b v all rights reserved

analytical and non analytical corrections to finite size scaling using conformal invariance the author calculates finite size corrections to the scaled spectra of one dimensional quantum chains at the critical point an explicit application to the ising and three state potts model gives good agreement with analytical and numerical results respectively

nonquadratic lyapunov functions for robust control abstract this paper deals with the problem of controlling a linear continuous time system with structured time varying parameter uncertainties and input disturbances with a lyapunov function approach in contrast with most of the previous results in the literature we do not confine our attention to the class of quadratic lyapunov functions conversely the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate lyapunov functions this question has a positive answer the proposed class is that of polyhedral norms or more generally of polyhedral minkowski functionals we show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a lyapunov function and a continuous state feedback compensator then it can be solved via a polyhedral lyapunov function and a possibly different continuous control moreover we show that the control can be piecewise linear a numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral

a new sum of squares design framework for robust control of polynomial fuzzy systems with uncertainties this paper presents a new sum of squares sos for brevity design framework for robust control of polynomial fuzzy systems with uncertainties two kinds of robust stabilization conditions are derived in terms of sos one is global sos robust stabilization conditions that guarantee the global and asymptotical stability of polynomial fuzzy control systems the other is semiglobal sos robust stabilization conditions the latter is available for very complicated systems that are difficult to guarantee the global and asymptotical stability of polynomial fuzzy control systems the main feature of all the sos robust stabilization conditions derived in this paper are to be expressed as nonconvex formulations with respect to polynomial lyapunov function parameters and polynomial feedback gains since a typical transformation from nonconvex sos design conditions to convex sos design conditions often results in some conservative issues the new design framework presented in this paper gives key ideas to avoid the conservative issues the first key idea is that we directly solve nonconvex sos design conditions without applying the typical transformation the second key idea is that we bring a so called copositivity concept these ideas provide some advantages in addition to relaxations to solve our sos robust stabilization conditions efficiently we introduce a gradient algorithm formulated as a minimizing optimization problem of the upper bound of the time derivative of an sos polynomial that can be regarded as a candidate of polynomial lyapunov functions three design examples are provided to illustrate the validity and applicability of the proposed design framework the examples demonstrate advantages of our new sos design framework for the existing linear matrix inequality approaches and the existing convex sos approach

conjugate convex lyapunov functions for dual linear differential inclusions tools from convex analysis are used to show how stability properties and lyapunov inequalities translate when passing from a linear differential inclusion ldi to its dual in particular it is proved that a convex positive definite function is a lyapunov function for an ldi if and only if its convex conjugate is a lyapunov function for the ldis dual examples show how such duality effectively doubles the number of tools available for assessing stability of ldis

critical behavior of the 3d ising model on a poissonian random lattice abstract the single cluster monte carlo algorithm and the reweighting technique are used to simulate the 3d ferromagnetic ising model on 3d voronoi delauney lattices it is assumed that the coupling factor j varies with the distance r between the first neighbors as j r e a r with a 0 the critical exponents and are calculated and according to the present estimates for the critical exponents we argue that this random system belongs to the same universality class of the pure 3d ferromagnetic ising model

brief homogeneous lyapunov functions for systems with structured uncertainties the problem addressed in this paper is the construction of homogeneous polynomial lyapunov functions hplfs for linear systems with time varying structured uncertainties a sufficient condition for the existence of an hplf of given degree is formulated in terms of a linear matrix inequalities lmi feasibility problem this condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the lyapunov function the maximum norm of the parametric uncertainty for which there exists a homogeneous polynomial lyapunov function is computed by solving a generalized eigenvalue problem the construction of such lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in lmi form comparisons with other classes of lyapunov functions through numerical examples taken from the literature show that hplfs are a powerful tool for robustness analysis

robust state feedback control of ltv systems nonlinear is better than linear examples of linear control systems with fast time varying uncertain coefficients are given which can be stabilized by a nonlinear memoryless state feedback but cannot be stabilized by a linear time invariant dynamic state feedback by means of one of these examples the authors show that the closed loop quadratic stability margin may be infinitely smaller than the actual stability margin

non self averaging in autocorrelations for potts models on quenched random gravity graphs we investigate the non self averaging properties of the dynamics of ising 4 state potts and 10 state potts models in single cluster monte carlo simulations on quenched ensembles of planar trivalent 3 random graphs which we use as an example of relevant quenched connectivity disorder we employ a novel application of scaling techniques to the cumulative probability distribution of the autocorrelation times for both the energy and magnetization in order to discern non self averaging although the specific results discussed here are for quenched random graphs the method has quite general applicability

brief on absolute stability analysis by polyhedral lyapunov functions this paper addresses the problem of computing a polyhedral lyapunov function plf for a linear differential inclusion ldi a numerically efficient algorithm is presented to compute a plf for an ldi it is also pointed out that for the case of a polyhedron defined by vertices rather than by faces one can state algebraic stability conditions analogous to those given by molchanov and pyatnitskiy 1986

absolute stability analysis of discrete time systems with composite quadratic lyapunov functions a generalized sector bounded by piecewise linear functions was introduced in a previous paper for the purpose of reducing conservatism in absolute stability analysis of systems with nonlinearity and or uncertainty this paper will further enhance absolute stability analysis by using the composite quadratic lyapunov function whose level set is the convex hull of a family of ellipsoids the absolute stability analysis will be approached by characterizing absolutely contractively invariant aci level sets of the composite quadratic lyapunov functions this objective will be achieved through three steps the first step transforms the problem of absolute stability analysis into one of stability analysis for an array of saturated linear systems the second step establishes stability conditions for linear difference inclusions and then for saturated linear systems the third step assembles all the conditions of stability for an array of saturated linear systems into a condition of absolute stability based on the conditions for absolute stability optimization problems are formulated for the estimation of the stability region numerical examples demonstrate that stability analysis results based on composite quadratic lyapunov functions improve significantly on what can be achieved with quadratic lyapunov functions

properties of composite laplacian quadratics and their applications in consensus of linear differential inclusions this technical note introduces a non quadratic function that is defined in terms of multiple laplacian quadratics called the function of composite laplacian quadratics the function is mainly used for control synthesis and analysis of multi agent systems some important properties of the function are studied in particular the one level set dissipation property is established and the gradient of the function is derived the function is further applied to solve the consensus problem of linear differential inclusions ldis finally a numerical example is given to verify the validity of the derived results

evidence of intermittent cascades from discrete hierarchical dissipation in turbulence abstract we present the results of a search of log periodic corrections to scaling in the moments of the square of velocity increments taken as proxies of the energy dissipation rate in experiments at high reynolds number 2500 of three dimensional fully developed turbulence a simple dynamical representation of the richardson kolmogorov cartoon of a cascade shows that standard averaging techniques erase by their very construction the possible existence of log periodic corrections to scaling associated with a discrete hierarchy to remedy this drawback we introduce a novel canonical averaging that we test extensively on synthetic examples constructed to mimic the interplay between a weak log periodic component and rather strong multiplicative and phase noises tests are also presented on 1 f5 3 random phase noise on the multifractal random walk model and on lattice and non lattice multifractal models our extensive tests confirm the remarkable observation of statistically significant log periodic corrections to scaling with a preferred scaling ratio for length scales compatible with the value 2 a strong confirmation of this result is provided by the identification of up to five harmonics of the fundamental log periodic undulations associated with up to five levels of the underlying hierarchical dynamical structure a natural interpretation of our results is that the richardson kolmogorov mental picture of a cascade becomes a realistic description if one allows for intermittent births and deaths of discrete cascades at varying scales

nonequilibrium relaxation study on spin glass model ferromagnetic fm transition point and universality class of the j ising model on cubic lattice are studied by computer simulation using the nonequilibrium relaxation ner method at fm bond density p between multicritical one pmc and 1 static exponents take universal values which are different from those of p 1 or p pmc and they are consistent with the values of the random fixed point and of the simulations and experiments for diluted ising systems the dynamical exponent however seems to depend on p and not to be universal although it may be due to finite time correction the estimation of pmc on square lattice is also refined to be 0 8894 9 which is consistent with the nishimori and nemoto s conjecture 0 889972

tests of conformal invariance in randomness induced second order phase transitions laboratoire de physique des mat eriaux universit e henri poincar e nancy 1 bo ite postale 239 f 54506 vand uvre les nancy cedex france received 6 july 1998 revised manuscript 29 september 1998 the conformal covariance of correlation functions is checked in the second order transition in duced by random bonds in the two dimensional eight state potts model the decay of correlationsis obtained via transfer matrix calculations in a cylinder geometry and large scale monte carlosimulations provide access to the correlations and the pro les inside a square with free or xedboundary conditions in both geometries conformal transformations constrain the form of the spa tial dependence leading to accurate determinations of the order parameter scaling index in goodagreement with previous independent determinations obtained through standard techniques theenergy density exponent is also computed pacs numbers 64 60 cn 05 50 q 05 70 jk 64 60 fr