coherent anomaly method in self avoiding walk problems abstract self avoiding walk saw being a nonequilibrium cooperative phenomenon is investigated with a finite order restricted walk finite orw or forw coherent anomaly method cam the coefficient 1 r in the asymptotic form c nr 1 r n 1 r for the total number c nr of r orw s with respect to the step number n is investigated for the first time an asymptotic form for saw s is thus obtained form the series of forw approximants c nr br g n 1 a r n as the envelope curve c n b ae g g n n g numerical results are given by c n 1 424 n 0 2788 4 1507 n and c n 1 179 n 0 1587 10 005 n for the plane triangular lattice and f c c lattice respectively a good coincidence of the total numbers estimated from the above simple formulae with exact enumerations for finite step saw s implies that the essential nature of saw non markov process can be understood from forw markov process in the cam framework

examination of the 1 d expansion method from exact enumeration for a self interacting self avoiding walk the 1 d expansion method for polymer chains is examined by comparing these expansions for several thermodynamic and structural quantities with the results of standard series analysis of exact enumeration data the comparisons cover a wide range of spatial dimensions d including non integer ones and are performed for particular values of interaction energy good agreement is generally found for d 4 whereas discrepancies become conspicuous as d decreases to d 2 reasonable values are obtained for the exponents v and gamma in d 2 4 by applying the coherent anomaly method of suzuki 1986 to our 1 d expansions through fifth order in d 1

decomposition formulas of exponential operators and lie exponentials with some applications to quantum mechanics and statistical physics decomposition formulas of general exponential operators in a banach algebra and in a lie algebra are presented that yield a basis of monte carlo simulation of quantum systems they are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena a global approximation method of transient phenomena is also formulated on the basis of decomposition formulas it is applied to the laser model as a simple example

sampled data h 2 optimization of systems with i o delays via analog loop shifting the problem of sampled data h2 control of systems with a single input delay is studied conventional solutions of sampled data optimal control problems begin with a conversion to a discrete time problem with input delays followed by a matching optimization method geared to handle delays we propose reversing that order and address the delay first by an analog loop transformation hence discretization and optimization involve no delays and are amenable to any standard method the proposed streamlined approach fits also previously unsolved problems where sampling and or hold functions are design parameters

system structure and singular control the general continuous time linear quadratic control problem is considered it is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem

construction and parameterization of all static and dynamic h 2 optimal state feedback solutions for discrete time systems abstract this paper considers an h 2 optimization problem via state feedback for discrete time systems the class of problems dealt with here has a left invertible transfer matrix function from the control input to the controlled output the paper constructs and parameterizes all the static and dynamic h 2 optimal state feedback solutions moreover all the eigenvalues of an optimal closed loop system are characterized all optimal closed loop systems share a set of eigenvalues which are termed the optimal fixed modes every h 2 optimal controller must assign among the closed loop eigenvalues the set of optimal fixed modes this set of optimal fixed modes includes a set of optimal fixed decoupling zeros which shows the minimum absolutely necessary number and locations of pole zero cancellations present in any h 2 optimal design most of the results presented here are analogous to but not quite the same as those for continuous time systems in fact there are some fundamental differences between the continuous and discrete time systems reflecting mainly the inherent nature and characteristics of these systems

1 d expansions for critical amplitudes the author shows how to expand critical amplitudes in inverse powers of the dimensionality d the technique which is rather general is applied to the classical n vector model with particular emphasis on the self avoiding walk n 0 and ising n 1 cases the existence of these expansions and of similar ones for critical points and critical constants suggests that 1 d expansions for non universal quantities are analogous to the epsilon expansions for universal quantities

1 d expansions for the free energy of weakly embedded site animal models of branched polymers we give the 1 d expansion to order 1 d5 for the limiting reduced free energy of a weakly embedded two variable site animal model of branched polymers hence we are able to derive expansions for the free energy of related one variable models and the growth constant of weakly and strongly embedded trees again to order 1 d5 we argue that the free energy expansions are asymptotically correct for a small range of fugacities only exact results show that animals with a compact hypercubic structure make the overwhelming contribution to the total number of weakly embedded site animals especially for large d this result is not reflected in the free energy 1 d expansions where trees make the dominant contribution we also derive new exact enumeration data for lattice animals using intermediate calculations in the derivation of the 1 d expansions thus we give partition functions for the general d dimensional hypercubic lattice for one to 13 sites

brief paper direct state space solution of multirate sampled data h2 optimal control in solving the multirate sampled data h 2 control problem using the lifting approach one needs to solve a constraint discrete time h 2 optimal control problem for a generalized plant with infinite dimensional input output spaces to solve this problem the existing sampled data h 2 design technique computes an equivalent finite dimensional discrete time system and then designs the optimal h 2 controller for the equivalent system in this paper we will show that this problem can be solved using state space formulas by dealing with operators directly the operator compositions are computed explicitly using discrete multirate lifting and matrix exponentials the advantages of the direct method are it is straightforward it has clear physical meanings and it is more efficient computationally a sufficient condition for the existence and uniqueness of multirate sampled data h 2 optimal controller is given in terms of the continuous time plant

on general ℋ2 control from frequency to time domain this article presents a general formula for discrete time 2 control it works with the regular and singular case of 2 control i e in the case of possibly non left invertible matrices g 12 and non right invertible matrices g 21 with possible unit circle invariant zeros in the generic case it can be simplified and adapted to work with the plant transfer matrix directly without invoking the matrices of the parameterisation of stabilising controllers a further result of this article is the presented necessary and sufficient conditions for state space 2 control under only stabilisability and detectability assumptions if the conditions are satisfied an observer based 2 controller is constructed the corresponding numerical algorithm consists of solving two discrete time algebraic riccati systems darss and two eigen problems

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

control under stochastic multiplicative uncertainties part ii optimal design for performance this paper studies the optimal control design problem for linear discrete time systems with stochastic multiplicative uncertainties these uncertainties are assumed to be present in the control inputs and modeled as independent and identically distributed i i d random processes the optimal performance under study is defined in the mean square sense referred to as the mean square optimal h 2 performance it is shown that the mean square optimal h 2 control problem via state feedback can be solved using a mean square stabilizing solution to a modified algebraic riccati equation mare a necessary and sufficient condition for the existence of this solution is presented which constitutes a generalization of the solution to the classic optimal h 2 state feedback design problem whereas the latter can be obtained by solving an algebraic riccati equation are it is also proven that the optimal control design problem can be cast as an eigenvalue problem evp for the output feedback case with possible input delays we show that the mean square optimal h 2 control problem also amounts to solving an mare when the plant has no nonminimum phase zeros from the inputs to the measurement outputs that is the global optimal solution is obtained by solving an mare incorporating the delays the implication is that in this case a separation principle still holds

from dilute to dense self avoiding walks on hypercubic lattices a field theoretic representation is presented to count the number of configurations of a single self avoiding walk on a hypercubic lattice ind dimensions with periodic boundary conditions we evaluate the connectivity constant as a function of the fractionf of sites occupied by the polymer chain the meanfield approximation is exact in the limit of infinite dimensions and corrections to it in powers ofd 1 can be systematically evaluated the connectivity constant and the site entropy calculated throughout second order compare well with known results in two and three dimensions we also find that the entropy per site develops a maximum atf 1 2d 1 ford 2 d 3 this maximum occurs atf 0 80 f 0 86 and its value is about 50 30 higher than the entropy per site of a hamiltonian walk f 1

h 2 and h infty design of sampled data systems using lifting part i general framework and solutions this paper presents a complete solution to the h2 and h infty problems for sampled data systems as opposed to previous works in the area it is assumed here that all or some of the sampling function the discrete time controller and the hold function are available for design the solution is obtained by transforming the problem to discrete time using the well known lifting technique it is then shown that the desired components of the sampled data controller can be peeled off from the inherently infinite dimensional description in the lifted domain the procedure for doing this last step is central to the approach in this paper both new and revised solutions are presented in this paper the solution to the h2 problems is completely new the solution to the h infty problems is presented in a unifying framework and is more transparent than the previous existing solutions in the literature transparency pays in the form of clearer results in particular a separation structure is established between the design of sub optimal hold and sampler

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

asymptotic properties of the best sampled data control performance properties of the best achievable performance of sampled data control systems are investigated contrary to our intuition the best sampled data control performance does not always converge to the best continuous time control performance even as the sampling period approaches zero a counterexample is presented and a lifting based analysis is performed it is shown that the performance convergence is guaranteed if and only if a certain hankel norm and an aliasing effect computed from a specified antialiasing filter tend to vanish with a decreasing sampling period it is also shown that this condition is reduced to a simpler form if the specified filter has special forms these results can be generalized to a system involving a generalized hold

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

on the zero freeness of tall multirate linear systems in this technical note tall discrete time linear systems with multirate outputs are studied in particular we focus on their zeros in systems and control literature zeros of multirate systems are defined as those of their corresponding time invariant systems obtained through blocking of the original multirate systems we assume that blocked systems are tall i e have more outputs than inputs it is demonstrated that for generic choice of the parameter matrices linear systems with multirate outputs generically have no finite nonzero zeros however they may have zeros at the origin or at infinity depending on the choice of blocking delay and the input state and output dimensions

universality in branched polymers on d dimensional hypercubic lattices the authors have derived series for weakly and strongly embeddable trees in d dimensional simple hypercubic lattices for arbitrary integral d for d 2 3 9 they present series evidence that such trees are in the same universality class as lattice animals in addition they have derived expansions in inverse powers of sigma 2d 1 for the growth parameters for bond and site trees and compare these with the corresponding results for animals

explicit expressions for cascade factorization of general nonminimum phase systems explicit expressions for two different cascade factorizations of any detectable left invertible nonminimum phase systems are given the first one is a well known minimum phase all pass factorization by which all nonminimum phase zeros of a transfer function g s are collected into an all pass factor v s and g s is written g sub m s v where g sub m s is considered as a minimum phase image of g s the second one is a new cascade factorization by which g s is rewritten as g sub m s u s where u s collects all awkward zeros including all nonminimum phase zeros of g s both g sub m s and g sub m s retain the given infinite zero structure of g s further properties of g sub m s g sub m s and u s are discussed these factorizations are useful in several applications including loop transfer recovery

scaling of self avoiding walks in high dimensions we examine self avoiding walks in dimensions 4 to 8 using high precision monte carlo simulations up to length n 16 384 providing the first such results in dimensions d 4 on which we concentrate our analysis we analyse the scaling behaviour of the partition function and the statistics of nearest neighbour contacts as well as the average geometric size of the walks and compare our results to 1 d expansions and to excellent rigorous bounds that exist in particular we obtain precise values for the connective constants 5 8 838 544 3 6 10 878 094 4 7 12 902 817 3 8 14 919 257 2 and give a revised estimate of 4 6 774 043 5 all of these are by at least one order of magnitude more accurate than those previously given from other approaches in d 4 and all approaches in d 4 our results are consistent with most theoretical predictions though in d 5 we find clear evidence of anomalous n 1 2 corrections for the scaling of the geometric size of the walks which we understand as a non analytic correction to scaling of the general form n 4 d 2 not present in pure gaussian random walks

high order non unitary split step decomposition of unitary operators we propose a high order numerical decomposition of exponentials of hermitian operators in terms of a product of exponentials of simple terms following an idea which has been pioneered by m suzuki implementing it for complex coefficients we outline a convenient fourth order formula which can be written compactly for an arbitrary number of non commuting terms in the hamiltonian and which is superior to the optimal formula with real coefficients both in complexity and accuracy we show asymptotic stability of our method for a sufficiently small time step and demonstrate its efficiency and accuracy in different numerical models

sampled data and discrete time h_2 optimal control this paper deals with the sampled data h 2 optimal control problem given a linear time invariant continuous time system the problem of minimizing the h 2 performance over all sampled data controllers with a fixed sampling period can be reduced to a pure discrete time h 2 optimal control problem this discrete time h 2 problem is always singular motivated by this in this paper we give a treatment of the discrete time h 2 optimal control problem in its full generality the results we obtain are then applied to the singular discrete time h 2 problem arising from the sampled data h 2 problem in particular we give conditions for the existence of optimal sampled data controllers we also show that the h 2 performance of a continuous time controller can always be recovered asymptotically by choosing the sampling period sufficiently small finally we show that the optimal sampled data h 2 performance converges to the continuous time optimal h 2 performance as the sampling period converges to zero

best achievable tracking performance in sampled data systems via lti controllers in this paper we study the problem of tracking a step reference signal using sampled data control systems we are interested in the tracking performance defined as the integral square of the tracking error response between the system s output and the reference input this performance is deemed the best achievable by a sampled data controller with a linear time invariant discrete time compensator if it is the minimal attainable by all such controllers that stabilize the system our primary objective is to investigate the fundamental tracking performance limit in sampled data systems and to understand whether and how sampling and hold in a sampled data system may impose intrinsic barriers to performance we consider two tracking performance measures with one defined with respect to the unit step signal and another with respect to a delayed step signal and averaged over one sampling period we derive an analytical closed form expression in each case for the best achievable performance the results show that a performance loss is generally incurred in a sampled data system in comparison to the tracking performance achievable by analog controllers this loss of performance as so demonstrated by the expressions is attributed to the non minimum phase behaviors as well as the intersample effects generated by samplers and hold devices thus sampled data controllers do result in an additional performance limit which is seen as a necessary tradeoff for other advantages offered by this class of controllers

new method of analysing self avoiding walks in four dimensions a new method based on the concept of fractal dimensionality is used to study the problem of self avoiding walks in a four dimensional lattice the authors find from monte carlo simulations that the confluent logarithmic exponent related to the end to end distance is 1 8 or 0 01 in excellent agreement with the prediction derived from the n to 0 vector model

general theory of fractal path integrals with applications to many body theories and statistical physics a general scheme of fractal decomposition of exponential operators is presented in any order m namely exp x a b sm x o xm 1 for any positive integer m where sm x et1a et2b et3a et4b etma with finite m depending on m a general recursive scheme of construction of tj is given explicitly it is proven that some of tj should be negative for m 3 and for any finite m nonexistence theorem of positive decomposition general systematic decomposition criterions based on a new type of time ordering are also formulated the decomposition exp x a b sm x n n o xm 1 nm yields a new efficient approach to quantum monte carlo simulations

existence of discrete time lqg controllers existence results for the lqg controller are investigated an infimal riccati equation based controller may potentially give closed loop eigenvalues on the unit circle assuming left and right invertibility it is shown that there exists an optimal controller if and only if the riccati equation based controller stabilizes the closed loop system after removal of all its unobservable and uncontrollable modes furthermore this reduced controller is the optimal controller and its transfer function is unique this existence condition is a considerable simplification of the more general geometric condition recently derived by trentelman and stoorvogel

new lattice model for interacting avoiding polymers with controlled length distribution a new lattice spin model for many self avoiding polymers is introduced in which the chain length distribution is fully controllable with a single generating magnetic field the model utilises spins with additional internal symmetry degrees of freedom to impose a causal connectivity of the polymer bonds on the lattice use of the method of random fields then produces an equivalent n to 0 limit field theory the flory huggins theory for a polymer solution emerges simply from this field theory in the mean field approximation polymer polymer interactions between polymer segments on nearest neighbour lattice are introduced into the field theory and the low polymer volume fraction limit of the theory reduces to the edwards type field theory for dilute through semidilute polymer solutions a sketch is provided towards the treatment of branched polymers with fully controllable chain and branch length distributions and branching probabilities as well as a kinetic polymerisation system governed by specified propagation and termination probabilities

novel system inversion algorithm with application to oversampled perfect reconstruction filter banks we derive a novel algorithm for linear discrete time system inversion with decision delay and frequency weighted norm criterion complexity of the algorithm grows only linearly with the decision delay the algorithm also applies to certain singular cases where optimal inverses may be nonunique in that case the set of optimal inverses is parametrized a scilab implementation of the algorithm is provided applications in oversampled perfect reconstruction filter banks are given

a bound of conservativeness in sampled data robust stabilization and its dependence on sampling periods abstract this paper investigates conservativeness of the l 2 induced norm in a sampled data robust stabilization problem namely it is shown that possible conservativeness of the l 2 induced norm can be bounded by difference between the best achievable performance of time invariant continuous time controllers and that of sampled data controllers moreover this difference is proven to approach zero as a sampling period tends to zero which means the conservativeness above loses importance when a sampling period is small these results are important because robust stabilization based on the l 2 induced norm is computationally inexpensive compared with robust stabilization based on the non conservative robust stability index one key lemma plays an essential role in this paper and it should be interesting independently

lattice trees with specified topologies the authors study the total numbers of lattice trees with specified topologies for strongly embeddable or site clusters with ni branching points of degree i they show how to prove rigorously that the growth constants exist and are all equal to the neighbour avoiding walk limit nu they derive some exact upper bounds for the critical exponents associated with the star comb and brush topologies exact enumeration data are derived and analysed for both weak and strong embeddings of some stars combs and brushes on the square triangular simple cubic and d 4 simple hypercubic lattices using the exact enumeration data for the general d dimensional simple hypercubic lattice and the exact results for the interior of a bethe lattice they derive expansions for the growth constants in inverse powers of the dimensionality these results are consistent with the growth constants being equal to the appropriate walk limits mu or nu

self avoiding walk contacts and random walk self intersections in variable dimensionality the average number of nearest neighbor nn contacts m of self avoiding walks saw s on a hypercubic lattice is calculated using direct enumeration and 1 d expansion methods where d is the spatial dimension these calculations are compared with exact analytic determinations for the asymptotic number of random walk rw self intersections in the limit of long chains n ensuremath rightarrow ensuremath infty the number of rw binary ternary etc self intersections is a function of the probability mathit c mathit d mathrm that a rw escapes from the origin to infinity and an accurate tabulation of mathit c mathit d mathrm is given in the dimension range 2d10 we find that the number of saw nn contacts m mathrm mathrm saw has an asymptotic behavior m mathrm mathrm saw ensuremath sim mathit a mathrm ensuremath infty n similar to that for the number of rw self intersections m mathrm mathrm rw ensuremath sim mathit a mathrm rw n as first suggested by domb but the corrections to this leading scaling differ for the rw and saw problems the contact amplitude mathit a mathrm ensuremath infty determined from direct enumeration data and ratio extrapolation exhibits a maximum near d 3 dimensions as does mathit a mathrm rw comparison of the numerical estimates for mathit a mathrm ensuremath infty to the 1 d expansion calculation of mathit a mathrm ensuremath infty shows significant deviation for d5 reflecting the strong fluctuations in contacts that arise in lower dimensions the correction to the scaling exponent mathrm ensuremath delta mathit m for saw nn contacts exhibits a maximum near d 2 dimensions a behavior similar to previous observations for the saw exponent ensuremath gamma estimates of the ensuremath theta point for interacting saw s the critical temperature of the o m model and other lattice constants e g bond and site percolation thresholds are obtained in terms of saw and rw lattice parameters

marriage of exact enumeration and 1 d expansion methods lattice model of dilute polymers we consider the properties of a self avoiding polymer chain with nearestneighbor contact energy on ad dimensional hypercubic lattice general theoretical arguments enable us to prescribe the exact analytic form of then segment chain partition functionc n and unknown coefficients for chains of up to 11 segments are determined using exact enumeration data ind 2 6 this exact form provides the main ingredient to produce a large n expansion ind 1of the chain free energy through fifth order with the full dependence on the contact energy retained the dependent chain connectivity constant and free energy amplitude are evaluated within thed 1expansion too d 5 our general formulation includes for the first time self avoiding walks neighboravoiding walks theta and collapsed chains as particular limiting cases

h2 optimal controllers with measurement feedback for continuous time systems flexibility in closed loop pole placement for a general h 2 optimal control problem at first all h 2 optimal measurement feedback controllers are characterized and parameterized and then attention is focused on controllers with observer based architecture both full order as well as reduced order observer based h 2 optimal controllers are characterized and parameterized also systematic methods of designing them are presented an important problem coined as an h 2 optimal control problem with simultaneous pole placement is formulated and solved that is since in general there exist many h 2 optimal measurement feedback controllers utilizing such flexibility and freedom we can solve the problem of simultaneously placing the closed loop poles at desirable locations whenever possible while still preserving h 2 optimality all the design algorithms developed here are easily computer implementable

hypercubic lattice saw exponents nu and gamma 3 99 dimensions revisited the self avoiding walk saw exponents nu and gamma are computed over a range of dimensions 1 or d infinity from exact expressions for the mean square end to end distance rn2 and the partition function qn of saws having a limited number of steps n or 11 saw exponents nu gamma for arbitrary dimension d are estimated by applying standard extrapolation techniques to the direct enumeration data which has been analytically continued to variable dimension exponent estimates obtained from continuum theories of self avoiding paths are compared with the saw calculations