design of state detectors for nonlinear systems using symmetries and semi invariants two techniques are proposed for the design of asymptotic state detectors for nonlinear systems both in the continuous time and discrete time cases the first technique allows one to design linear state detectors for nonlinear systems belonging to a given class which is studied and characterized in the paper the second technique extends the ideas of the first by using semi invariants to compute invariant sets and study their stability

on polynomial vector fields having a given affine variety as attractive and invariant set application to robotics the main goal of this paper is to compute a class of polynomial vector fields whose associated dynamical system has a given affine variety as attractive and invariant set a given point in such an affine variety as invariant and attractive and another given affine variety as invariant set solving the application of this technique in the robotic area this objective is reached by using some tools taken from algebraic geometry practical examples of how these vector fields can be computed are reported moreover by using these techniques two feedback control laws respectively for a unicycle like mobile robot and for a car like mobile robot which make them move within the workspace approaching to a selected algebraic curve are given

critical behaviour of a fluid in a random shear flow renormalization group analysis of a simplified model critical behaviour of a fluid binary mixture or liquid crystal subjected to strongly anisotropic turbulent mixing is studied by means of the field theoretic renormalization group as a simplified model relaxational stochastic dynamics of a non conserved scalar order parameter coupled to a random velocity field with prescribed statistics is considered the velocity is taken gaussian white in time with a correlation function of the form t t k d where k is the component of the wave vector perpendicular to the distinguished direction direction of the flow the d dimensional generalization of the ensemble introduced by avellaneda and majda 1990 commun math phys 131 381 within the context of passive scalar advection it is shown that depending on the relation between the exponent and the space dimensionality d the system exhibits various types of large scale self similar behaviour associated with different infrared attractive fixed points of the renormalization group equations in addition to well known asymptotic regimes model a of equilibrium critical dynamics and a passively advected scalar with no self interaction the existence of a new non equilibrium and strongly anisotropic type of critical behaviour universality class is established and the corresponding critical dimensions are calculated to the second order of the double expansion in and e 4 d two loop approximation the most realistic values of the model parameters for example d 3 and the kolmogorov exponent 4 3 belong to this class the scaling behaviour appears anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different the results are in qualitative agreement with the results obtained in experiments and simulations of fluid systems subjected to various kinds of regular and chaotic anisotropic flows

on a lyapunov equation for polynomial continuous time systems the primary goal of this paper is to describe the classes of all polynomial vector fields such that the associated dynamical system admits some polynomial lyapunov function v x by fixing or constraining the time derivative of v x in different ways several different cases are studied hints are given about how to use the results for stability analysis and for non polynomial systems

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

semi invariants and their use for stability analysis of planar systems semi invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvector eigenvalue pair for linear systems and are therefore very useful to depict the behaviour of the system in this article semi invariants are used to construct explicitly lyapunov functions useful for studying the stability of the origin for continuous time systems moreover using well known tools from differential geometry such as orbital symmetries it is shown how semi invariants can be found for several classes of systems important connections with centre manifold theory are pointed out by using the proposed general techniques a new proof of a result claimed to bendixson bendixson i 1901 sur les courbes definies par des equations differentielles acta mathematics 24 1 88 is given

dynamic scaling of disordered ising systems the critical dynamics of site disordered three dimensional ising systems has been studied by conventional local dynamics simulations investing 1015 single spin flips the author has verified that the correlation functions of disordered ising systems obey dynamic scaling behaviour the dynamic scaling functions have been calculated the critical exponent z appears to be strongly dependent on the degree of disorder the author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour the dynamical exponent has been estimated to be z 2 4 or 0 1 which is surprisingly near to the renormalization group result in o in 1 2

local and cluster critical dynamics of the 3d random site ising model abstract we present the results of monte carlo simulations for the critical dynamics of the three dimensional site diluted quenched ising model three different dynamics are considered these correspond to the local update metropolis scheme as well as to the swendsen wang and wolff cluster algorithms the lattice sizes of l 10 96 are analysed by a finite size scaling technique the site dilution concentration p 0 85 was chosen to minimize the correction to scaling effects we calculate numerical values of the dynamical critical exponents for the integrated and exponential autocorrelation times for energy and magnetization as expected cluster algorithms are characterized by lower values of dynamical critical exponent than the local one also in the case of dilution critical slowing down is more pronounced for the metropolis algorithm however the striking feature of our estimates is that they suggest that dilution leads to decrease of the dynamical critical exponent for the cluster algorithms this phenomenon is quite opposite to the local dynamics where dilution enhances critical slowing down

a flexible condition number for weighted linear least squares problem and its statistical estimation in this paper we mainly focus on the derivation of a flexible condition number for weighted linear least squares problem and its estimation with frechet derivative and kronecker product the explicit expression of the condition number is obtained when the coefficient matrices are large and dense considering the difficulties of explicitly forming the expression of condition number in computer we also present its simplified form through a simple and easy to operate method two different condition estimation methods are introduced and the numerical experiments are also performed to show the efficiency of the condition estimators

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

on the ferromagnetic ising model in noninteger spatial dimension using the necessary and sufficient conditions in terms of the high field series coefficients that the yang lee theorem holds we prove rigorously by counterexample that it cannot be extended to general noninteger dimension when such models are defined by the natural analytic continuation

critical crossover phenomena in disordered ising systems the author has investigated the critical behaviour of site disordered ising systems over a large range of dilution 0 6 or p or 1 0 investing about 1015 single spin flips with conventional local dynamics critical temperatures have been determined with high precision using the cumulant method the finite size simulations lead to concentration dependent exponents which seem to violate universality the author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region the author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder

deformations for linear periodic discrete time systems the adjoint normal form following the technique proposed by arnold for matrices time invariant autonomous systems and extended by tannenbaum to time invariant linear control systems with inputs and outputs a normal form for linear periodic discrete time systems depending on physical parameters is proposed the considered actions are deformations i e linear periodic transformations depending on the parameters being identity at the nominal values of them

crossover effects in dilute magnetic materials by finite time dynamics method abstract crossover phenomena are ubiquitous in disordered system however the crossover effects from the competition between different fixed points make it hard to identify the asymptotic scaling regime controlled by the random fixed point therefore the analyses on the crossover effects are of great importance to understand phase transition in dilute magnetic materials in this paper we derive the scaling function characterizing the crossovers to probe critical behavior according to these we show an alternative method to locate the asymptotic critical region in the quenched random systems in the finite time dynamics frame for the three dimensional bond diluted ising model the asymptotic critical exponents are identified with 0 686 1 0 356 6 0 057 3 1 345 20 z 2 170 11 at middle bond concentration p 0 7 from the effective ones describing the approach to the asymptotic regime for weak dilution p 0 9 and strong dilution p 0 5 the dynamic critical exponent from our non equilibrium simulation supports distinctly z 2 18 as suggested previously as opposed to the existing larger values

critical dynamics of the three dimensional ising model a monte carlo study extensive monte carlo simulations have been performed to analyze the dynamical behavior of the three dimensional ising model with local dynamics we have studied the equilibrium correlation functions and the power spectral densities of odd and even observables the exponential relaxation times have been calculated in the asymptotic one exponential time region we find that the critical exponentz 2 09 0 02 characterizes the algebraic divergence with lattice size for all observables the influence of scaling corrections has been analyzed we have determined integrated relaxation times as well their dynamical exponentzint agrees withz for correlations of the magnetization and its absolute value but it is different for energy correlations we have applied a scaling method to analyze the behavior of the correlation functions this method verifies excellent scaling behavior and yields a dynamical exponentzscal which perfectly agrees withz

model c critical dynamics of disordered magnets the critical dynamics of model c in the presence of disorder is considered it is known that in the asymptotics a conserved secondary density decouples from the nonconserved order parameter for disordered systems however couplings between order parameter and secondary density cause considerable effects on non asymptotic critical properties here a general procedure for a renormalization group treatment is proposed already the one loop approximation gives a qualitatively correct picture of the diluted model c dynamical criticality a more quantitative description is achieved using two loop approximation in order to get reliable results resummation technique has to be applied

exact and approximate feedback linearization without the linear controllability assumption the feedback linearization problem of nonlinear control systems has been solved in the literature under the assumption that the nonlinear system is linearly controllable in this paper the assumption of linear controllability is removed necessary and sufficient conditions are given through lie orbital symmetries thus giving a geometric characterization of the problem in the analytic case both the exact and the approximate linearization problems are considered in the analytic case

effects of mixing and stirring on the critical behaviour stochastic dynamics of a nonconserved scalar order parameter near its critical point subject to random stirring and mixing is studied using the field theoretic renormalization group the stirring and mixing are modelled by a random external gaussian noise with the correlation function t t k4 d y and the divergence free due to incompressibility velocity field governed by the stochastic navier stokes equation with a random gaussian force with the correlation function depending on the relations between the exponents y and y and the space dimensionality d the model reveals several types of scaling regimes some of them are well known model a of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow but there are three new non equilibrium regimes universality classes associated with new nontrivial fixed points of the renormalization group equations the corresponding critical dimensions are calculated in the two loop approximation second order of the triple expansion in y y and e 4 d

on the crossover to universal criticality in dilute ising systems monte carlo simulations of the critical behaviour of disordered ising systems have led to concentration dependent critical exponents which seem to violate universality we apply the renormalization group to investigate the crossover effects which cause the observed behaviour we improve the three loop expansion of the callan symanzik and wilson functions by a pade borel approximation and solve the flow equations for various initial points the exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point

stabilization of polynomial nonlinear systems by algebraic geometry techniques this technical note applies methods from algebraic geometry namely polynomial ideals and sub modules to find parameterized sets of state feedback stabilizing control laws for polynomial continuous time affine nonlinear systems the proposed method is illustrated by simple examples

inverse problems for invariant algebraic curves explicit computations given an algebraic curve in the complex affine plane we describe how to determine all planar polynomial vector fields which leave this curve invariant if all finite singular points of the curve are non degenerate we give an explicit expression for these vector fields in the general setting we provide an algorithmic approach and as an alternative we discuss sigma processes

critical properties of ising models with long range interactions on sierpinski gasket lattices in higher euclidean dimensions abstract ising systems with two spin long range interactions are studied on sierpinski gasket lattices generalized to euclidean dimensions d 3 4 5 6 for each lattice the long range interactions are assigned to edges that are part of the appropriate self similar structure in consequence these systems are infinitely ramified and exhibit critical behavior at finite temperatures an analysis of critical properties of the systems in particular the calculation of critical exponents is carried out by using an exact renormalization group method it is shown that there exists some similarity between critical properties of the long range interacting fractal models under study here and critical properties of short range interacting ising models defined on abstract regular lattices interpolated to noninteger dimensionalities

the condition numbers for weighted moore penrose inverse and weighted linear least squares problem condition numbers play an important role in numerical analysis classical normise condition numbers are used to measure the size of both input perturbations and output errors in this paper we study the weighted normwise relative condition numbers for the weighted moore penrose inverse and the weighted linear least squares wls problems in the case of the full column rank matrix the bounds or formulas for the weighted condition numbers are presented the obtained results can be viewed as extensions of the earlier works studied by others

critical exponents of the dilute ising model from four loop expansions four loop expansions of the coefficients of the callan symanzik equation for 3d and 2d dilute ising models are calculated fixed point coordinates and critical exponents are estimated by two methods of summation of double series a generalisation of the pade borel approximation and the first confluent form of the e algorithm of wynn summation of the double series for the 2d dilute ising model gives exponents very close to exponents of the pure ising model in accordance with the exact solution

compilation of two point and four point graphs in field theory in noninteger dimensions two point and four point graphs appearing in a three loop approximation in the field theory renormalization group scheme are calculated in general dimensions combining feynman parameterization and direct integration loop integrals are represented in the form of the expressions depending on the space dimension d as a parameter the expressions obtained enables one to consider renormalization group functions directly at noninteger d

condition numbers and perturbation of the weighted moore penrose inverse and weighted linear least squares problem it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of nonsingular linear systems here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b these explicit expressions extend the earlier work of several authors as we know the computed weighted least squares solution x of min email protected email protected m in the presence of the round off error satisfies the perturbed equation min email protected a e y b f m finally an upper bound of email protected n is derived for the case where the weighted matrix and vector norms and the assumption rank a e rank a are used

a contribution to the theory of condition in this paper the use of the condition number of a problem as defined by rice in 1966 is discussed for the eigenvalue eigenvector and linear least squares problems either condition numbers according to various norms are determined or lower and upper bounds for them are derived

critical dynamics a field theoretical approach we review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of halperin and hohenberg we unify the presentation of the theoretical background by restricting ourselves to the field theoretic renormalization group method the main results obtained in the different universality classes are presented this contains the critical dynamics near the gas liquid transition in pure fluids model h the plait point and consolute point in mixtures model h the superfluid transition in 4 he model f and 4 he 3 he mixtures model f the curie point model j and neel point model g in heisenberg magnets and the superconducting transition in comparison with experimental results it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non universal effective behaviour either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics e g at the superfluid transition or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas liquid or demixing transition the calculation of the critical dynamics is adequate in most cases only in two loop order we review these results and present the solution to unreasonable features found for some models thus we consider model c where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model model a have not been understood in general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role their fixed point values and the approach to the fixed point are changed considerably in two loop order compared to their values in one loop order these considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium model f and the isotropic antiferromagnet model g as far as possible the comparison of results obtained by the renormalization group theory with numerical simulations has been made

two loop feynman integrals for ϕ4 theory with long range correlated disorder two loop massive feynman integrals for 4 field theoretical model with long range correlated disorder are considered massive integrals for the vertex function 4 including two or three massless propagators for generic space dimension and for any value of the correlation parameter are evaluated analytically applying mellin barnes method as well as familiar representation for one loop integrals obtained expressions are presented in the form of hypergeometric functions

switching signal estimator design for a class of elementary systems the goal of this technical note is to design a switching signal estimator for a class of elementary continuous time switching or switched systems first the elementary system is recast into a polynomial form and secondly some tools borrowed from algebraic geometry are used to express the switching signal as a function of the time derivatives of the output and of the input

damage spreading and critical exponents for model a ising dynamics using damage spreading and heat bath dynamics we study the ising model in 2 and 3 dimensions with non conservative dynamics our algorithm differs in some important points from previous ones which makes it rather efficient we give estimates for the exponent z which seem to be the most precise published so far 2 172 0 006 for d 2 2 032 0 004 for d 3 we also give precise estimates of the exponent introduced by janssen et al z phys b 73 1989 539 and of analogous but in principle independent exponents we find surprisingly that some of the latter agree with and give an explanation for this

critical exponents of random ising like systems in general dimensions critical exponents of weakly dilute ising like systems are computed for non integer space dimensionalities in the range 2 d 4 the calculations are performed in the framework of the callan symanzik field theoretic approach two loop renormalization group functions are obtained as renormalized perturbation theory series expansions directly in noninteger dimensions the values of the critical exponents are estimated with the use of the two variable borel resummation method

brief paper stability analysis of planar systems with nilpotent non zero linear part a simple necessary and sufficient condition for asymptotic stability of the origin is derived for a large class of planar nonlinear systems that cannot be studied by means of their linearization which is nilpotent and not zero the condition is based on the use of darboux polynomials to derive a lyapunov function for the first approximation of the system with respect to a vector function representing a dilation results by andreev 1958 on the qualitative behaviour of equilibrium points and the belitskii normal form are used in the proofs the proposed condition is applied to the asymptotic stabilization of the origin for a class of planar systems whose linearization cannot be stabilized

small sample statistical condition estimates for general matrix functions a new condition estimation procedure for general matrix functions is presented that accurately gauges sensitivity by measuring the effect of random perturbations at the point of evaluation in this procedure the number of extra function evaluations used to evaluate the condition estimate determines the order of the estimate that is the probability that the estimate is off by a given factor is inversely proportional to the factor raised to the order of the method the transpose free nature of this new method allows it to be applied to a broad range of problems in which the function maps between spaces of different dimensions this is in sharp contrast to the more common power method condition estimation procedure that is limited in the usual case where the frechet derivative is known only implicitly to maps between spaces of equal dimension a group of examples illustrates the flexibility of the new estimation procedure in handling a variety of problems and types of sensitivity estimates such as mix

on the critical fluctuations in superconductors the field theoretical gauge model for a superconductor generalized to an n 2 component complex order parameter is considered the question of the order of the phase transition occurring in this model is discussed previous renormalization group calculations suggest that in such a model a fluctuation induced first order phase transition occurs we re examine previously obtained expressions for the renormalization group functions in the two loop approximation in three dimensions special attention is being paid to the fact that the corresponding series might be the asymptotic ones and therefore have zero radius of convergence we discuss the possible ways of analytical continuation of the series obtained comparing results obtained by direct calculations with those obtained by pade analysis and the pade borel resummation technique the conjecture is made that in the model under consideration there still exists a possibility for a second order phase transition

on the corrections to scaling in three dimensional ising models the leading correction to scaling amplitudes for the spin 1 2 nearest neighbor sc bcc and fee ising models are considered with the particular aim of determining their signs on the basis of previous two variable series analyses by chen fisher and nickel and renormalization group 4 d expansions it is concluded that the correction amplitudes for the susceptibility correlation length specific heat and spontaneous magnetization arenegative for all three lattices thus for example the effective exponent eff t asymptotically approaches the true susceptibility exponent fromabove other earlier and more recent high temperature series and field theoretic analyses are seen to be consistent with this result however the usual nonasymptotic perturbative field theoretic approaches are essentially committed to positive correction amplitudes the question of the signs therefore relates directly to the applicability of these non asymptotic field theoretic calculations to three dimensional ising models as well as to different experimental systems

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

statistical condition estimation for linear systems the standard approach to measuring the condition of a linear system compresses all sensitivity information into one number thus a loss of information can occur in situations in which the standard condition number with respect to inversion does not accurately reflect the actual sensitivity of a solution or particular entries of a solution it is shown that a new method for estimating the sensitivity of linear systems addresses these difficulties the new procedure measures the effects on the solution of small random changes in the input data and by properly scaling the results obtains reliable condition estimates for each entry of the computed solution moreover this approach which is referred to as small sample statistical condition estimation is no more costly than the standard 1 norm or power method 2 norm condition estimates and it has the advantage of considerable flexibility for example it easily accommodates restrictions on or structure associated with allowable perturbations the method also has a rigorous statistical theory available for the probability of accuracy of the condition estimates however it gives no estimate of an approximate null vector for nearly singular systems the theory of this approach is discussed along with several illustrative examples

sign of corrections to scaling amplitudes field theoretic considerations and results for self repelling walks i discuss the application of renormalized field theory to such critical systems where the renormalized coupling approaches the fixed point from the wrong strong coupling side in contrast to a belief sometimes expressed in the literature i find that also this situation can be described within the standard formalism only the interpretation of the strong coupling branch relies on the existence of a finite cutoff i illustrate these considerations with an analysis of monte carlo data for self avoiding walks finding that the data support the theory in all respects however it is stressed that the nonuniversal parameters of the renormalized theory in general have no simple relation to the parameters of the underlying bare model