the pro nilpotent group topology on a free group abstract in this paper we study the pro nilpotent group topology on a free group first we describe the closure of the product of finitely many finitely generated subgroups of a free group in the pro nilpotent group topology and then present an algorithm to compute it we deduce that the nil closure of a rational subset of a free group is an effectively constructible rational subset and hence has decidable membership we also prove that the g nil kernel of a finite monoid is computable and hence pseudovarieties of the form v g nil have decidable membership problem for every decidable pseudovariety of monoids v finally we prove that the semidirect product j g nil has a decidable membership problem

expansions free inverse semigroups and schutzenberger product in this paper we shall present a new construction of the free inverse monoid on a set x contrary to the previous constructions of 9 11 our construction is symmetric and originates from classical ideas of language theory the ingredients of this construction are the free group on x and the relation that associates to a word w of the free monoid on x the set of all pairs u v such that uv w it follows at once from our construction that the free inverse monoid on x can be naturally embedded into the schutzenberger product of two free groups of basis x we shall also give some connections with the theory of expansions as developed by rhodes and birget 2 3

the uniform word problem for groups and finite rees quotients of e unitary inverse semigroups abstract if c is a class of groups closed under taking subgroups we show that the decidability of the uniform word problem for c is implied by the decidability of the membership problem for the class of finite rees quotients of e unitary inverse semigroups with maximal group image in c the converse is shown if c is a pseudovariety when c is a pseudovariety the above problems are shown to be equivalent to the problem of embedding a finite labeled graph in the cayley graph of a group in c this latter problem is shown to be equivalent to deciding whether a finite labeled graph is a schutzenberger graph of an e unitary inverse semigroup with maximal group image in c

on the complexity of computing the profinite closure of a rational language profinite topology is used in the classification of rational languages in particular several important decidability problems related to the malcev product reduce to the computation of the closure of a rational language in the profinite topology it is known that given a rational language by a deterministic automaton computing a deterministic automaton accepting its profinite closure can be done with an exponential upper bound this paper is dedicated the study of a lower bound for this problem we prove that in some cases if the alphabet contains at least three letters it requires an exponential time

analytic stability margin design for unstable and nonminimum phase plants this note is concerned with the problem of designing con trollers for unstable and nonminimum phase plants according to stability margin criteria first an elementary technique to achieve a satisfactory tradeoff between phase and gain margin is presented then it is shown how to derive from the controller thus obtained other controllers that make ei ther margin almost twice as large the procedures perform well in terms of both computational complexity and controller order

topology optimization of sheets in contact by a subgradient method we consider the solution of finite element discretized optimum sheet problems by an iterative algorithm the problem is that of maximizing the stiffness of a sheet subject to constraints on the admissible designs and unilateral contact conditions on the displacements the model allows for zero design volumes and thus constitutes a true topology optimization problem we propose and evaluate a subgradient optimization algorithm for a reformulation into a non differentiable convex minimization problem in the displacement variables the convergence of the method and its low computational complexity are established an optimal design is derived through a simple averaging scheme which combines the solutions to the linear design problems solved within the subgradient method to illustrate the efficiency of the algorithm and investigate the properties of the optimal designs thealgorithm is numerically tested on some medium and large scale problems 1997 by john wiley sons ltd

first and second order methods for semidefinite programming in this paper we survey the most recent methods that have been developed for the solution of semidefinite programs we first concentrate on the methods that have been primarily motivated by the interior point ip algorithms for linear programming putting special emphasis in the class of primal dual path following algorithms we also survey methods that have been developed for solving large scale sdp problems these include first order nonlinear programming nlp methods and more specialized path following ip methods which use the preconditioned conjugate gradient or residual scheme to compute the newton direction and the notion of matrix completion to exploit data sparsity

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

simultaneous partial pole placement a new approach to multimode system design simultaneous partial pole placement of a family of single input single output plants is proposed as a generalization of the classical pole placement and stabilization problems this problem finds application in the design of a compensator for a family of linear dynamical systems in this note we show that the proposed problem is equivalent to a new class of transcendental problem using stable minimum phase rational functions with real coefficients a necessary condition for the solvability of the associated transcendental problem is obtained finally a counterexample to the following conjecture is obtained pairs of simultaneously stabilizable plants of bounded mcmillan degree have simultaneously stabilizing compensators of bounded mcmillan degree

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

the geometry of profinite graphs with applications to free groups and finite monoids we initiate the study of the class of profinite graphs defined by the following geometric property for any two vertices v and w of there is a unique smallest connected profinite subgraph of containing them such graphs are called tree like profinite trees in the sense of gildenhuys and ribes are tree like but the converse is not true a profinite group is then said to be dendral if it has a tree like cayley graph with respect to some generating set a bass serre type characterization of dendral groups is provided also such groups including free profinite groups are shown to enjoy a certain small cancellation condition we define a pseudovariety of groups h to be arboreous if all finitely generated free pro h groups are dendral with respect to a free generating set our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids we prove for arboreous pseudovarieties h a pro h analog of the ribes and zalesskii product theorem for the profinite topology on a free group also arboreous pseudovarieties are characterized as precisely the solutions h to the much studied pseudovariety equation j h j h

potential reduction polynomial time method for truss topology design the multiload truss topology design problem is modeled as a minimization of the maximum with respect to k loading scenarios compliance subject to equilibium constraints and restrictions on the bar volumes the problem may involve a very large number of potential bars so as to allow a rich variety of topologies the original formulation is in terms of two sets of variables m bar volumes and n nodal displacements several equivalent convex reformulations of this problem are presented these convex problems although highly nonlinear possess nice analytical structure and therefore can be solved by an interior point potential reduction method associated with appropriate logarithmic barrier for the feasible domain of the problem for this method to improve the accuracy of the current a proximate solution by an absolute constant factor it suffices in the worst case to perform o sqrt km newton steps with o k 3 n 3 k 2 n 2 m operations per step

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

analytic phase margin design doyle et al 1992 presented an algorithm for analytic phase margin control design without special care however the compensator computed with this algorithm may not be a real rational function the problem is evident when the plant has real unstable poles in this case the algorithm requires a mapping of real points into complex values and it is not clear that the resulting compensator has real coefficients the purpose of this paper is to show how a complex mapping required in this algorithm can always be selected so that the compensator does have real coefficients

a constructive version of the ribes zalesski product theorem for any given finitely generated subgroups h1 hn of a free group f and any element w of f not contained in the product h1 hn a finite quotient of f is explicitly constructed which separates the element w from the set h1 hn this provides a constructive version of the product theorem stating that h1 hn is closed in the profinite topology of f the method of proof also applies to other profinite topologies it is efficient for the profinite topology as well as for the pro p topology of f the main tools used are universal p extensions and inverse automata

on polyhedral approximations of the second order cone we demonstrate that a conic quadratic problem is polynomially reducible to linear programming we demonstrate this by constructing for every an lp program explicitly given in terms of e and the data of cqp with the following properties i the number dimx dimu of variables and the number dimp of constraints in lp do not exceed ii every feasible solutionx to cqp can be extended to a feasible solution x u to lp iii if x u is feasible for lp thenx satisfies the e relaxed constraints of cqp namely

modified nevanlinna pick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor abstract this paper is a sequel to an earlier work in which we applied classical novnnlinna pick interpolation to the stabilization of a single input output plant with uncertainly in the gain factor by means of an asymptotically stable control and feedback sensor in this present work we show that for the proper design we need a modification of the classical interpolation procedure a point not clearly brought out in our previous paper

design of planar articulated mechanisms using branch and bound this paper considers an optimization model and a solution method for the design of two dimensional mechanical mechanisms the mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously the underlying mechanical analysis model is based on a truss representation allowing for large displacements for mechanisms undergoing large displacements elastic stability is of major concern we derive conditions modeled by nonlinear matrix inequalities which guarantee that a stable equilibrium is found and that buckling is prevented the feasible set of the design problem is described by nonlinear differentiable and non differentiable constraints as well as nonlinear matrix inequalities to solve the mechanism design problem a branch and bound method based on convex relaxations is developed to guarantee convergence of the method two different types of convex relaxations are derived the relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub problems encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

strong duality for semidefinite programming it is well known that the duality theory for linear programming lp is powerful and elegant and lies behind algorithms such as simplex and interior point methods however the standard lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps semidefinite linear programming sdp is a generalization of lp where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables there are many applications e g in systems and control theory and combinatorial optimization however the lagrangian dual for sdp can have a duality gap we discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming these duals guarantee strong duality i e a zero duality gap and dual attainment this paper is motivated by the recent paper by ramana where one of these duals is introduced

h optimal feedback controllers for linear multivariable systems this paper treats the problem of designing for a linear multivariable plant a feedback controller which minimizes the h infty norm of a weighted sensitivity matrix there exists a family of optimal improper feedbacks this family is determined by application of a theory of ball and helton a method for computing optimal feedbacks is described in detail and a numerical example is included it is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices

on h optimal sensitivity theory for siso feedback systems this paper deals with the design of feedback controllers which minimize the h infty norm of the sensitivity function suitably weighted this approach to the theory of feedback design was introduced by zames 1 and developed by zames and francis 2 in this paper the theory of sarason 3 is applied to the determination of the optimal weighted sensitivity function and an upper bound on its norm the problem of achieving small sensitivity over a specified frequency band is studied and the effect of nonminimum phase is elucidated finally a method is introduced for handling plant poles and zeros on the imaginary axis

multivariable feedback sensitivity and decentralized control in this paper the problem of sensitivity reduction by feedback is studied and related to a problem of decentralized control a plant will be represented by an n times n matrix of frequency responses which may be unstable or irrational the object will be to find conditions on p s under which a diagonal feedback f s can make the sensitivity parallel i p s f s 1 parallel arbitrarily small over some specified frequency interval j omega 0 j omega 0 without violating a global sensitivity bound parallel i p s f s 1 parallel leq m m geq some const 1 for re s geq 0 it will be shown that such a diagonal feedback of the high gain type can be constructed whenever p 1 s is analytic in re s geq 0 p s satisfies an attenuation condition near s infty and p s approaches diagonal dominance at high frequencies it will also be shown that these conditions on the plant can be interpreted as conditions for the existence of a decentralized wide band control scheme

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

constructing divisions into power groups the result due to henckell margolis pin and rhodes modulo ash s solution to the pointlike conjecture that every finite block group divides a power group has long been considered to be one of the deepest results in finite semigroup and algebraic automata theory however the proof is not constructive solving a long standing problem we provide in this paper an explicit construction of such a division we also generalize the result to a large class of pseudovariefies of groups local group pseudovarieties are also considered generalizing and making constructive results of margolis and the second author some applications to language theory are mentioned

varieties of finite supersolvable groups with the m hall property the varieties in the title are shown to be precisely the product varieties gp ab d for some prime p and some positive integer d dividing p 1 here gp denotes the variety of all finite p groups and ab d the variety of all finite abelian groups of exponent dividing d it turns out that these are exactly those varieties h of supersolvable groups for which all finitely generated free pro h groups are freely indexed in the sense of lubotzky and van den dries several alternative characterizations of these varieties are presented some applications to formal language theory and finite monoid theory are also given among these is the determination of all supersolvable solutions h to the equations ph j h and j h jopen image in new windowh which is to the present date the most complete solution to a problem raised by pin another consequence of our results is that for each such variety h the monoid variety ph j h jopen image in new windowh has decidable membership

new models for optimal truss topology in limit design based on unified elastic plastic analysis abstract this paper presents several equivalent formulations for a structural design problem where the load carrying capacity is maximized for a prescribed volume subject to bounds on complementary energy and stresses this limit design model covers the full range of strictly elastic elastic plastic and strictly plastic designs and is based on the unified analysis model of ben tal and taylor 1 while this design model is convex it is nonlinearly constrained and is of a very high dimension for topology design problems application of duality principles leads to several simpler but nonsmooth equivalent models in particular for the case when the design variables do not have explicit bounds the dual models reduce to a minimization subject to a single linear constraint of a pointwise maximum of a finite number of convex functions more importantly these simpler design models are of greatly reduced size since they contain only nodal variables further the two cases of strictly plastic as well as strictly elastic limit design models can be reduced to linear programs both of which unexpectedly are shown to be equivalent to the more widely studied model for minimum compliance topology design of elastic trusses several numerical examples illustrate the usefulness of these new dual formulations

consistency of open loop experimental frequency response data with coprime factor plant models the model data consistency problem for coprime factorization considered is given some possibly noisy frequency response data obtained by running open loop experiments on a system show that these data are consistent with a given family of perturbed coprime factor models and a time domain spl lscr sub spl infin noise model in the noise free open loop case the model data consistency problem boils down to the existence of an interpolating function in spl rscr spl hscr sub spl infin that evaluates to a finite number of complex matrices at a finite number of points on the imaginary axis a theorem on boundary interpolation in spl rscr spl hscr sub spl infin is a building block that allows one to devise computationally simple necessary and sufficient tests to check if the perturbed coprime factorization is consistent with the data for standard coprime factorizations the test involves the computation of minimum norm solutions to underdetermined complex matrix equations the schmidt mirsky theorem is used in the case of special factorizations of flexible systems for spl lscr sub spl infin noise corrupting the frequency response measurements a complete solution to the open loop noisy siso problem using the structured singular value spl mu is given

a convex parameterization of robustly stabilizing controllers this paper treats synthesis of robust controllers for linear time invariant systems uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial the main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations with the new parameterization certain robust performance problems can be stated in terms of quasi convex optimization

correction to optimal control of perturbed linear static systems this paper presents a typographical correction to the proof of theorem 4 in the above mentioned paper smith et al 1996

non euclidian metrics and the robust stabilization of systems with parameter uncertainty this paper considers from a complex function theoretic point of view certain kinds of robust synthesis problems in particular we use a certain kind of metric on the disk the hyperbolic metric which allows us to reduce the problem of robust stabilization of systems with many types of real and complex parameter variations to an easily solvable problem in non euclidian geometry it is shown that several apparently different problems can be treated in a unified general framework a new result on the gain margin problem for multivariable plants is also given finally we apply our methods to systems with real zero or pole variations

a note on robust stabilization for systems with parameters we consider the problem of stabilizing a linear time invariant multivariable system whose transfer function admits a factorization which contains parametric type uncertainties the parameters are known to take values in some region a we first give a sufficient condition for robust stability when a proper feedback compensator is used here it is important to note that no assumption is made about the number of unstable plant poles following this we focus attention on how to compute an appropriate robustly stabilizing compensator this is done in a two step procedure initially the solutions of a diophantine equation generate a class of proper compensators each of which stabilizes the nominal closed loop system this is followed by a nyquist theorem inspired minimization which yields the final choice an illustrative physical example is given the compensator thus constructed guarantees stability over u a discussion about how to compute the size of the attainable robust stability region u is also included

robust stabilization some extensions of the gain margin maximization problem considers robust stabilization for siso systems with linear dependence on a real uncertain parameter which may occur either in the numerator or denominator of the plant transfer function the authors show necessary and sufficient conditions for the existence of robust controllers their structure and parameterization can be obtained from the corresponding nevanlinna pick interpolation problems this work extends the previously known case of an uncertain gain parameter