on the slip boundary condition for liquid flow over planar porous boundaries navier stokes flow within the pores of and over a porous body is averaged using scale dependent weighting function methodology equations governing steady uniaxial averaged flow parallel to a planar porous boundary are derived and used to examine the status of the ad hoc slip boundary condition of beavers and joseph particular attention is drawn to the roles played by the averaging length scale and behaviour in the associated boundary interfacial region

finite approximations of markov operators in this paper we survey some recent developments in the numerical analysis of markov operators and in particular frobenius perron operators associated with chaotic discrete dynamical systems

analysis of linear structured systems using a primal dual algorithm in this paper we give a polynomial algorithm to compute the infinite structure of a structured system a directed graph is associated with such structured systems the infinite zero orders can be computed on the associated graph via the determination of the minimal length of vertex disjoint input output paths this search corresponds to a minimum cost flow determination on an appropriate directed graph the proposed algorithm is based on the primal dual algorithm linked to linear programming this polynomial algorithm is one of the most efficient for this type of problem moreover it allows an iterative determination of the generic infinite structure which is a key tool for solving numerous control problems

discretization of the frobenius perron operator using a sparse haar tensor basis the sparse ulam method the global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated frobenius perron operator for systems with low dimensional long term dynamics efficient techniques exist for a numerical approximation of the most important eigenfunctions cf m dellnitz and o junge siam j numer anal 36 1999 pp 491 515 they are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor ulam s method in this paper we develop a numerical technique which makes ulam s approach applicable to systems with higher dimensional long term dynamics it is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids c zenger sparse grids in parallel algorithms for partial differential equations kiel 1990 vieweg braunschweig 1991 pp 241 251 h j bungartz and m griebel acta numer 13 2004 pp 147 269 here we use a sparse haar tensor basis as the underlying approximation space we develop the technique establish statements about its complexity and convergence and present two numerical examples

error estimates of the markov finite approximation of the frobenius perron operator in the practical application of the modern ergodic theory one of the main problems is related to the computation of the absolutely continuous invariant measures for nonsingular transformations on measure spaces see lasota and mackey l a simple example is constructed in li 2 to show that the round off error can completely dominate the calculation if one is to compute the invariant measure directly using a sequence of cesaro sums an alternative is to compute the invariant density function for the frobenius perron operator p l 0 1 l 0 1 associated with the nonsingular measurable transformations s 0 11 0 11 the frobenius perron operator p is defined by

comments on achieving diagonal interactor matrix for multivariable linear systems with uncertain parameters abstract in this note we propose a criterion for determining whether or not a multivariable linear system may be made to have a diagonal interactor with the possible use of diagonal precompensation we present a constructive procedure to achieve a diagonal interactor for multivariable linear systems when possible this procedure is simpler than that given in gibbens et al automatica 29 1547 1550 1993

achieving diagonal interactor matrix for multivariable linear systems with uncertain parameters abstract the notion of interactor matrix or equivalently the hermite normal form is a generalization of relative degree to multivariable systems and is crucial in problems such as decoupling inverse dynamics and adaptive control in order for a system to be input output decoupled using static state feedback the existence of a diagonal interactor matrix must first be established for a multivariable linear system which does not have a diagonal interactor matrix dynamic precompensation or dynamic state feedback is required for achieving a diagonal interactor matrix for the compensated system such precompensation often depends on the parameters of system and is thus difficult to implement with accuracy when the system is subject to parameter uncertainty in this paper we characterize a class of linear systems which can be precompensated to achieve a diagonal interactor matrix without the exact knowledge of the system parameters more precisely we present necessary and sufficient conditions on the transfer matrix of the system under which there exists a diagonal dynamic precompensator such that the compensated system has a diagonal interactor matrix these conditions are associated with the so called non generic singularity of certain matrix related to the system structure but independent of the system parameters the result of this paper is expected to be useful in robust and adaptive designs

modified brinkman equation for a free flow problem at the interface of porous surfaces the cantor taylor brush configuration case 1 the free flow problem above at the surface interface and inside a cantor taylor brush configuration ctb simulating a porous medium was studied particle image velocimetry piv measurements confirm that the microscale stokes equation provides an accurate solution to the ctb microscale flow problem a comparison between the results of the averaged microscale stokes equation and that of the brinkman equation using an apparent viscosity shows that the concept of apparent viscosity did not provide a satisfactory agreement between the two approaches in order to develop a description of the average velocity profile across the interface flow region theoretical and numerical analyses were performed an averaging procedure of the navier stokes equations provided a set of three equations which were used to predict the average velocity in the fluid phase u f this set of equations is the suggested modified brinkman equation mbe the comparison between the results of the stokes equation and the mbe shows that optimizing the size of the averaging representative volume provides a good fit between the flow problem and the solution of the modified brinkman equation

computing invariant measures of piecewise convex transformations lets 0 1 0 1 be a piecewise convex transformation satisfying some conditions which guarantee the existence of an absolutely continuous invariant probability measure we prove the convergence of a class of markov finite approximations for computing the invariant measure using a compactness argument forl1 spaces

on the approximation of invariant measures given a discrete dynamical system defined by the map x x the density of the absolutely continuous a c invariant measure if it exists is the fixed point of the frobenius perron operator defined on l1 x ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator t y li proved the convergence of the scheme for expanding maps of the interval g keller and m blank extended this result to piecewise expanding maps of the cube in n we show convergence of a variation of ulam s scheme for maps of the cube for which the frobenius perron operator is quasicompact we also give sufficient conditions on for the existence of a unique fixed point of the matrix approximation and if the fixed point of the operator is a function of bounded variation we estimate the convergence rate

prior information in the design of multivariable adaptive controllers this note discusses the design of a dynamic precompensator w c s for an m times m mimo plant w p s such that for almost all cases w p s w c s has a diagonal hermite form

structured systems within the transfer matrix approach application to decoupling abstract we consider a new approach to structured systems within the transfer matrix framework we associate with such a structured system a graph which concentrates both knowledge on structural properties of independent subsystems and on their interconnections the analysis of this associated graph allows to conclude on the decouplability of structured systems

decoupling of structured systems by parameter independent precompensation and state feedback in this paper structured systems described by state space models are considered for these systems the entries of the state space model matrices are assumed to be either fixed zeros or free independent parameters we study the dynamic decoupling of structured linear systems it turns out that a large class of systems which is not state feedback decouplable can be made decouplable by the addition of a simple structured precompensator such a precompensator then has the property to be independent of the system parameters we give a necessary and sufficient condition for a structured system to be made feedback decouplable using a parameter independent precompensator furthermore we prove that when such a solution exists the precompensator can be chosen to be diagonal in this case the precompensator is constructed via a simple combinatorial algorithm our approach is based on a graph representation of the structured system all the results of this paper are related to some particular shortest sets of input output paths in the graph associated with the structured system

on the structure at infinity of linear square decoupled systems the purpose of this note is to demonstrate that a square invertible linear system c a b is decouplable by static state feedback if and only if its infinite zero orders are respectively equal to the infinite zero orders of the subsystems c i a b where c i denotes the i th row of c until now only the necessity of this condition has been established 1

a maximum entropy method for solving frobenius perron operator equations let s 0 1 0 1 be a nonsingular transformation that preserves an absolutely continuous invariant measure with density f in this paper we develop a general algorithm for approximating f using the maximum entropy approach

continuum equations of balance via weighted averages of microscopic quantities exact continuum forms of balance for mass momentum and energy are established as relations between weighted space time averages of molecular quantities computed at any supra atomic length time scales the choice of weighting func tion and the physical interpretation of all terms are discussed

markov finite approximation of frobenius perron operator the problem of existence and computation of absolutely continuous invariant measures for nonsingular transformations on measure spaces is one of the main concerns in the modern ergodic theory 3 lasota and yorke have established the existence of the invariant measures for a class of nonsingular measurable transformations s from 0 l to itself 4 later in 6 li and yorke gave a sufficient condition for the uniqueness of this invariant density and thus for the ergodicity of the mapping the classical birkhoff s individual ergodic theorem says that if u is an ergodic invariant probability measure under s then for any measurable set a c 0 11 the time average lim 1 n c ax s x which measures the average time spent in a under iteration of s exists to be i for p almost all x where xa is the characteristic function of a 1 on a and 0 off a this seems to suggest that we may use the cesaro sum of the above form to calculate the invariant measure a simple and important example in 5 however shows that the computer round off error can completely dominate the calculation and make the implementation difficult in order to overcome this difficulty li 5 proposed a rigorous numerical procedure which can be implemented on a computer with negligible round off error problem a mapping s 0 l 0 l is called piecewise c2 if there is a partition 0 a a a 1 of 0 l such that for k 1 r the restriction sk of s on uk a is a c2 function which can be extended to the closed interval ak i ak as a c2 function s need not be continuous at the point ak for a c 0 11 we write s a for x s x e a throughout the paper we assume s is a nonsingular measurable transformation i e s is measurable and for any measurable subset a c 0 l with m a 0 m s a 0 here m denotes the lebesgue measure on 0 11 the operator ps l 0 1 l 0 1 defined by