random copolymers and the morita approximation polymer adsorption and polymer localization we analyse directed walk models of random copolymer adsorption and localization ideally we would like t os olve thequenched problem but it appears to be intractable even for simple directed models the annealed approximation is solvable but is inadequate in the strong interaction regime fo rt he adsorption problem and gives a qualitatively incorrect phase diagram for the localization problem in this paper we treat these directed models using an approximation suggested by morita 1964 j math phys 5 1401 5 in which the proportion of each comonomer is fixed we find that the morita approximation leads to behaviour that is closer to that of the quenched average model and this is particularly interesting in the localization problem where the phase diagram is a tl east qualitatively very similar to that of the quenched average problem we also show that the phase boundaries in the morita approximation are bounds on the locations of the phase boundaries of the quenched model

a class of implicit peer methods for stiff systems we present a class of s stage implicit two step peer methods for the solution of stiff differential equations using the function values from the previous step in addition to the new function values this allows to increase the order to p s and to ensure zero stability straightforwardly corresponding s stage methods for s 6 of order p s with optimal zero stability are presented and their stability is discussed under special conditions we prove that an optimally zero stable subclass of these methods is superconvergent of order p s 1 for variable step sizes numerical tests and comparison with ode15s show the high potential of this class of implicit peer methods

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

brief paper on observability of nonlinear steady state generators an output regulation problem can be converted into a stabilization problem in a systematic framework thanks to an appropriately designed internal model however the successful construction of an internal model im relies on the observability condition of a steady state generator ssg in the literature the observability condition is restricted to linear or linearized ssgs this paper introduces a novel observability condition for nonlinear ssgs nonlinear observability leads to an effective combination of the internal model principle and the certainty equivalence principle for a more general class of systems

strong stability preserving explicit peer methods in this paper we study explicit peer methods up to order p 13 which have the strong stability preserving ssp property this class of general linear methods has the favourable property of a high stage order the effective ssp coefficient is maximized by solving a nonlinear constraint optimization problem numerically to high precision the coefficient matrices of the optimized methods are sparse in a very structured way linear multistep methods are obtained as a special case of only one stage

on the existence of a kazantzis kravaris luenberger observer we state sufficient conditions for the existence on a given open set of the extension to nonlinear systems of the luenberger observer as it has been proposed by kazantzis and kravaris we prove it is sufficient to choose the dimension of the system giving the observer less than or equal to 2 twice the dimension of the state to be observed we show that it is sufficient to know only an approximation of the solution of a pde needed for the implementation we establish a link with high gain observers finally we extend our results to systems satisfying an unboundedness observability property

parallel two step w methods with peer variables a new class of methods for the solution of stiff initial value problems is introduced that is parallel by design it has a two step character and propagates s different peer solution variables with essentially identical characteristics from step to step the main work lies in the solution of s independent linear stage equations which may be solved in parallel convergence of order s 1 and stability for general stepsize sequences are proved conditions for order s and stronger stability criteria are addressed as well promising methods up to order 7 are identified by numerical tests with some widely used stiff test problems some of these are competitive with existing software even in sequential computations

peer two step methods with embedded sensitivity approximation for parameter dependent odes peer two step methods have been successfully applied to initial value problems for stiff and nonstiff ordinary differential equations odes both on parallel and sequential computers their essential property is the use of several stages per time step with the same accuracy as a new application area these methods are now used for parameter dependent odes where the peer stages approximate the solution also at different places in the parameter space the main interest here is sensitivity data through an approximation of solution derivatives in different parameter directions basic stability and convergence properties are discussed and peer methods of order 2 and 3 in the time stepsize are constructed the computed sensitivity matrix is used in approximate newton and gauss newton methods for shooting in boundary value problems where initial values and or ode parameters are searched for and in parameter identification from partial information on trajectories

distributed output regulation of nonlinear multi agent systems via host internal model this note presents a host internal model h im approach to distributed output regulation of reaching nonlinear leader following consensus focusing on a basic nonlinear scenario of identical or homogeneous agents of a normal form it is shown that the problem is solvable as long as its interaction digraph contains a certain directed spanning tree a constructive lyapunov protocol is proposed by incorporating a single h im the result is also demonstrated by fitzhugh nagumo fhn and lorenz type nonlinear dynamic networks

doubly quasi consistent parallel explicit peer methods with built in global error estimation recently kulikov presented the idea of double quasi consistency which facilitates global error estimation and control considerably more precisely a local error control implemented in such methods plays a part of global error control at the same time however kulikov studied only nordsieck formulas and proved that there exists no doubly quasi consistent scheme among those methods here we prove that the class of doubly quasi consistent formulas is not empty and present the first example of such sort this scheme belongs to the family of superconvergent explicit two step peer methods constructed by weiner schmitt podhaisky and jebens we present a sample of s stage doubly quasi consistent parallel explicit peer methods of order s 1 when s 3 the notion of embedded formulas is utilized to evaluate efficiently the local error of the constructed doubly quasi consistent peer method and hence its global error at the same time numerical examples of this paper confirm clearly that the usual local error control implemented in doubly quasi consistent numerical integration techniques is capable of producing numerical solutions for user supplied accuracy conditions in automatic mode

a remark on the problem of semiglobal nonlinear output regulation this paper shows how the method for semiglobal stabilization by output feedback based on high gain observer and saturation and the method for structurally stable regulation based on system immersion can be effectively coupled in order to solve problems of robust output regulation in the presence of parameter uncertainties ranging over compact sets

reprint of peer methods with improved embedded sensitivities for parameter dependent odes recently an extension of peer two step methods to parameter dependent ivps has been proposed by the author and e kostina where additional embedded stages allow cheap computation of solution derivatives sensitivities with respect to problem parameters for each parameter only one additional satellite stage was used which provides a high order approximation to a neighboring solution however the accuracy of the sensitivities was only first order in the time stepsize h in the present paper we derive implicit peer methods where the sensitivity accuracy is of second order with the same number of stages approximate implementations yield efficient linearly implicit methods for stiff problems and explicit predictor corrector type versions for nonstiff problems tests confirm improved and more robust convergence of inexact newton iterations in shooting for different applications with nonstiff and stiff ivps

uniform practical nonlinear output regulation in this paper we present a solution to the problem of asymptotic and practical semiglobal regulation by output feedback for nonlinear systems a key feature of the proposed approach is that practical regulation is achieved uniformly with respect to the dimension of the internal model and to the gain of the stabilizer near the zero error manifold this property renders the approach interesting for a number of real cases by bridging the gap between output regulation theory and advanced engineering applications simulation results regarding meaningful control problems are also presented

on the localization transition of random copolymers near selective interfaces in this note we consider the de localization transition for random directed 1 1 dimensional copolymers in the proximity of an interface separating selective solvents we derive a rigorous lower bound on the free energy this yields a substantial improvement on the bounds from below on the critical line that were known so far our result implies that the critical curve does not lie below the critical curve set forth by monthus eur phys j b 13 111 130 2000 on the base of a renormalization group analysis we discuss this result in the light of the rigorous and non rigorous approaches present in the literature and by making an analogy with a particular asymptotics of a disordered wetting model we propose a simplified framework in which the question of identifying the critical curve as well as understanding the nature of the transition may be approached

peer methods with improved embedded sensitivities for parameter dependent odes recently an extension of peer two step methods to parameter dependent ivps has been proposed by the author and e kostina where additional embedded stages allow cheap computation of solution derivatives sensitivities with respect to problem parameters for each parameter only one additional satellite stage was used which provides a high order approximation to a neighboring solution however the accuracy of the sensitivities was only first order in the time stepsize h in the present paper we derive implicit peer methods where the sensitivity accuracy is of second order with the same number of stages approximate implementations yield efficient linearly implicit methods for stiff problems and explicit predictor corrector type versions for nonstiff problems tests confirm improved and more robust convergence of inexact newton iterations in shooting for different applications with nonstiff and stiff ivps

explicit two step peer methods we present a new class of explicit two step peer methods for the solution of nonstiff differential systems a construction principle for methods of order p s s the number of stages with optimal zero stability is given two methods of order p 6 found by numerical search are tested in matlab on several representative nonstiff problems the comparison with ode45 confirms the high potential of the new class of methods

copolymers at selective interfaces new bounds on the phase diagram we investigate the phase diagram of disordered copolymers at the interface between two selective solvents and in particular its weak coupling behavior encoded in the slope mc of the critical line at the origin we focus on the directed walk case which has turned out to be in spite of the apparent simplicity extremely challenging in mathematical terms the partition function of such a model does not depend on all the details of the markov chain that models the polymer but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns this observation leads to a natural generalization of the model in terms of arbitrary laws of return times the most interesting case being the one of return times with power law tails with exponent 1 1 2 in the case of the symmetric random walk the main results we present here are

order conditions for general linear methods we describe the derivation of order conditions without restrictions on stage order for general linear methods for ordinary differential equations this derivation is based on the extension of the albrecht approach proposed in the context of runge kutta and composite and linear cyclic methods this approach was generalized by jackiewicz and tracogna to two step runge kutta methods by jackiewicz and vermiglio to general linear methods with external stages of different orders and by garrappa to some classes of runge kutta methods for volterra integral equations with weakly singular kernels this leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods nordsieck methods and general linear methods with inherent runge kutta stability exact coefficients for several low order methods with some desirable stability properties are presented for illustration

the order of the localization transition for a random copolymer we use monte carlo methods and a scaling analysis to estimate the crossover exponent for a model of the localization transition for a random copolymer at the interface between two immiscible liquids the model that we use is a directed walk in two dimensions we find evidence that the nature of the phase transition is different in different regions of the phase diagram this agrees with previous results for a self avoiding walk model we also use exact enumeration and series analysis methods to estimate the locations of the phase boundaries and compare our results to a rigorous bound obtained by partial annealing

a general class of two step runge kutta methods for ordinary differential equations a general class of two step runge kutta methods that depend on stage values at two consecutive steps is studied these methods are special cases of general linear methods introduced by butcher and are quite efficient with respect to the number of function evaluations required for a given order general order conditions are derived using the approach proposed recently by albrecht and examples of methods are given up to the order 5 these methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments

output stabilization via nonlinear luenberger observers the present paper addresses the problem of the existence of an output feedback law that asymptotically steers to zero prescribed outputs while keeping all state variables bounded for any initial conditions in a given compact set the problem can be viewed as an extension of the classical problem of semiglobally stabilizing the trajectories of a controlled system to a compact set the problem also encompasses a version of the classical problem of output regulation under only a weak minimum phase assumption it is shown that there exists a controller solving the problem at hand the paper is deliberately focused on theoretical results regarding the existence of such a controller practical aspects involving the design and the implementation of the controller are left to a forthcoming work

nonlinear observer design using lyapunov s auxiliary theorem abstract the present work proposes a new approach to the nonlinear observer design problem based on the early ideas that influenced the development of the linear luenberger observer theory the proposed approach develops a nonlinear analogue the formulation of the observer design problem is realized via a system of singular first order linear pdes and a rather general set of necessary and sufficient conditions for solvability is derived by using lyapunov s auxiliary theorem the solution to the above system of pdes is locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package

local and global error estimation and control within explicit two step peer triples this paper elaborates a global error estimation and control mechanism in explicit two step peer methods these recently designed methods exhibit their high efficiency even in comparison to the best explicit runge kutta pairs more precisely we form here triples of the so called superconvergent explicit peer schemes and show that they are cheap and able to achieve preassigned accuracy conditions in automatic mode for comparison we present also numerical data derived by built in explicit matlab ode solvers implemented with only local error control especially we point out that a scaled global error is computed and regulated in this paper in contrast to the earlier published results where the absolute values of the global error have been utilized

search for efficient general linear methods for ordinary differential equations we analyze implicit general linear methods with s internal stages and r s 1 external stages of order p s 1 and stage order q s or q s 1 these methods might eventually lead to more efficient formulas than the class of dimsims and the class of general linear methods with inherent runge kutta stability we analyze also error propagation and estimation of local discretization errors examples of such methods which are a and l stable are derived up to the stage order q 3 or q 4 and order p 4

explicit multi step peer methods for special second order differential equations abstract the construction of s stage explicit two and three step peer methods of order p 2 s and p 3 s is considered for the solution of non stiff second order initial value problems where the right hand side does not depend on y by numerical search with respect to large stability intervals and small error constants sequential and parallel methods are constructed numerical tests of these peer methods in matlab and comparisons with a runge kutta nystrom method show the efficiency of the proposed methods

an adaptive newton picard algorithm with subspace iteration for computing periodic solutions this paper is concerned with the efficient computation of periodic orbits in large scale dynamical systems that arise after spatial discretization of partial differential equations pdes a hybrid newton picard scheme based on the shooting method is derived which in its simplest form is the recursive projection method rpm of shroff and keller siam j numer anal 30 1993 pp 1099 1120 and is used to compute and determine the stability of both stable and unstable periodic orbits the number of time integrations needed to obtain a solution is shown to be determined only by the system s dynamics this contrasts with traditional approaches based on newton s method for which the number of time integrations grows with the order of the spatial discretization two test examples are given to show the performance of the methods and to illustrate various theoretical points

localization transition for a polymer near an interface consider the directed process i s i where the second component is simple random walk on mathbb z s 0 0 define a transformed path measure by weighting each n step path with a factor exp lambda sum 1 leq i leq n omega i h sign s i here omega i i geq 1 is an i i d sequence of random variables taking values pm 1 with probability 1 2 acting as a random medium while lambda in 0 infty and h in 0 1 are parameters the weight factor has a tendency to pull the path towards the horizontal because it favors the combinations s i 0 omega i 1 and s i 0 omega i 1 the transformed path measure describes a heteropolymer consisting of hydrophylic and hydrophobic monomers near an oil water interface we study the free energy of this model as n to infty and show that there is a critical curve lambda to h c lambda where a phase transition occurs between localized and delocalized behavior in the vertical direction we derive several properties of this curve in particular its behavior for lambda downarrow 0 to obtain this behavior we prove that as lambda h downarrow 0 the free energy scales to its brownian motion analogue