finite monoids and the fine structure of nc 1 recently a new connection was discovered between the parallel complexity class nc 1 and the theory of finite automata in the work of barrington on bounded width branching programs there nonuniform nc 1 was characterized as those languages recognized by a certain nonuniform version of a dfa here we extend this characterization to show that the internal structures of nc 1 and the class of automata are closely related in particular using therien s classification of finite monoids we give new characterizations of the classes ac 0 depth k ac 0 and acc the last being the ac 0 closure of the mod q functions for all constant q we settle some of the open questions in 3 give a new proof that the dot depth hierarchy of algebraic automata theory is infinite 8 and offer a new framework for understanding the internal structure of nc 1

efficient algorithms for membership in boolean hierarchies of regular languages this paper provides efficient algorithms that decide membership for classes of several boolean hierarchies for which efficiency or even decidability were previously not known we develop new forbidden chain characterizations for the single levels of these hierarchies and obtain the following results the classes of the boolean hierarchy over level 1 of the dot depth hierarchy are decidable in nl previously only the decidability was known the same remains true if predicates mod d for fixed d are allowed if modular predicates for arbitrary d are allowed then the classes of the boolean hierarchy over level 1 are decidable for the restricted case of a two letter alphabet the classes of the boolean hierarchy over level 2 of the straubing therien hierarchy are decidable in nl this is the first decidability result for this hierarchy the membership problems for all mentioned boolean hierarchy classes are logspace many one hard for nl the membership problems for quasi aperiodic languages and for d quasi aperiodic languages are logspace many one complete for pspace

finite semigroup varieties defined by programs abstract we study the regular languages recognized by polynomial length programs over finite semigroups belonging to product varieties v li where v is a variety of finite monoids and li is the variety of finite locally trivial semigroups in the case where the semigroup variety has a particular closure property with respect to programs we are able to give precise characterizations of these regular languages as a corollary we obtain new proofs of the results of barrington compton straubing and therien characterizing the regular languages in certain circuit complexity classes

multiple time scalings of a multi output observer form time scalings in the multi output observer form for uncontrolled nonlinear continuous time systems are considered in this technical note it is the multi output version of an existing single input result time scaling broadens the class of systems which admits an exact error linearization observer design by including time scaling transformations the existence conditions of the time scaling transformation and the change of state coordinates to time scaled observer form are provided

non linear observer design by observer canonical forms this paper studies the non linear observer design problem by observer canonical forms we present necessary and sufficient conditions in terms of the ranks of matrices for the existence of a non linear state coordinate change under which a time variable system is transformed into an observer canonical form these conditions give the multiple versions of li and tao 1986 for the single output case then a comparison is made with the lie algebraic conditions obtained by krener and respondek 1985 and modified by the authors in a recent paper xia and gao 1987

observer linearization by output dependent time scale transformations we study the problem of observer linearization for single output dynamical systems in the presence of output dependent time scaling changes an alternative algorithm for the solution of the observer linearization problem is introduced the algorithm employs an exterior calculus approach that provides a simple procedure for the solution of the observer linearization problem by means of an output dependent time scale transformation

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

bennequin s inequality and the positivity of the signature we use an algorithm for special diagrams to prove a bennequin type inequality tor the signature of an arbitrary link diagram related to its murasugi sum decomposition we apply this inequality to show that the signature of a non trivial positive 3 braid knot is greater than its genus and that the signature of a positive braid link is minorated by an increasing function of its negated euler characteristic the latter property is conjectured to extend to positive links

programs over semigroups of dot depth one abstract the notion of a p variety arises in the algebraic approach to boolean circuit complexity it has great significance since many known and conjectured lower bounds on circuits are equivalent to the assertion that certain classes of semigroups form p varieties in this paper we prove that semigroups of dot depth one form a p variety this example has the following implication if a boolean combination of 1 formulas using arbitrary numerical predicates defines a regular language one can then find an equivalent 1 formula all of whose numerical predicates are regular

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

extended high gain observer design for a class of mimo non uniformly observable systems abstract this paper deals with the design of an extended high gain observer ehgo for a large class of multiple input multiple output mimo non uniformly observable systems indeed the involved system nonlinearities does not assume the usual triangular structure as it is often the case when a high gain approach is adopted this structure constraint relaxation is achieved by an appropriate use of the characteristic indices associated to the state components there are two fundamental features that are worth to be pointed out firstly the gain of the proposed ehgo is provided by an appropriate riccati ordinary differential equation ode secondly the convergence of the observation error is guaranteed under some persistent excitation condition which can be checked on line simulation results are given to illustrate the performance of the proposed ehgo

linearization by generalized input output injection abstract the problem under interest is the linearization of nonlinear mimo systems by generalized input output injection in order to design observers with linear error dynamics the method is based on the study of the structure of the input output differential equations thus the problem is solved as a realization problem in this note one considers the linearization under two kinds of inout output injection in the first case the transformation depends on the output and time derivatives of the input whereas in the second case derivatives of both the input and the output are considered necessary and sufficient conditions are obtained which generalize the ones on standard input output injection linearization

nonlinear observer design by observer error linearization this paper studies the observer design problem by the observer error linearization approach for nonlinear systems with and without inputs necessary and sufficient conditions for the existence of the linearization transformation are derived for nonlinear systems without inputs the conditions are shown to be corrections to an existing result a computation procedure and a different set of necessary and sufficient conditions based on the computation procedure are presented

canonical form observer design for non linear time variable systems an observer of canonical phase variable form for non linear time variable systems is introduced the development of this non linear time variable form requires regularity of the non linear time variable observability matrix of the system from the relationships derived during the development it follows that a non linear time variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non linearities about the reconstructed state trajectories is permissible this is an assumption similar to that of the extended kalman filter based on a linearization about the current estimate

languages defined with modular counting quantifiers we prove that a regular language defined by a boolean combination of generalized 1 sentences built using modular counting quantifiers can be defined by a boolean combination of 1 sentences in which only regular numerical predicates appear the same statement with 1 replaced by first order is equivalent to the conjecture that the nonuniform circuit complexity class acc is strictly contained in nc1 the argument introduces some new techniques based on a combination of semigroup theory and ramsey theory which may shed some light on the general case

circuit bottom fan in and computational power we investigate the relationship between circuit bottom fan in and circuit size when circuit depth is fixed we show that in order to compute certain functions a moderate reduction in circuit bottom fan in will cause significant increase in circuit size in particular we prove that there are functions that are computable by circuits of linear size and depth k with bottom fan in 2 but require exponential size for circuits of depth k with bottom fan in 1 a general scheme is established to study the trade off between circuit bottom fan in and circuit size based on this scheme we are able to prove for example that for any integer c there are functions that are computable by circuits of linear size and depth k with bottom fan in o log n but that require exponential size for circuits of depth k with bottom fan in c and that for any constant epsilon 0 there are functions that are computable by circuits of linear size and depth k with bottom fan in log n but that require superpolynomial size for circuits of depth k with bottom fan in o log 1 epsilon n a consequence of these results is that the three input read modes of alternating turing machines proposed in the literature are all distinct

non uniform automata over groups abstract a new model non uniform deterministic finite automata nudfa s over general finite monoids has recently been developed as a strong link between the theory of finite automata and low level parallel complexity achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non uniform nc1 nudfa characterizations of several important subclasses of nc1 and a new proof of the old result that the dot dephth hierarchy is infinite using m sipser s 1983 in proceedings 15th acm symposium on the theory of computing association for computing machinery new york pp 61 69 work on constant depth circuits here we extend this theory to nudfa s over solvable groups nudfa s over non solvable groups have the maximum possible computing power we characterize the power of nudfa s over nilpotent groups and prove some optimal lower bounds for nudfa s over certain groups which are solvable but not nilpotent most of these results appeared in preliminary form in d a barrington and d therien 1987 in automata languages and programming 14th international colloquium springer verlag berlin pp 163 173

counting alternating knots by genus it is shown that the number of alternating knots of given genus g 1 grows as a polynomial of degree 6g 4 in the crossing number the leading coefficient of the polynomial which depends on the parity of the crossing number is related to planar trivalent graphs with a bieulerian path the rate of growth of the number of such graphs is estimated

observer with linear error dynamics for nonlinear multi output systems based upon nonlinear state coordinate transformations and a condensed dual brunovsk form for multi output systems a method for designing nonlinear state observers with linear error dynamics is presented necessary and sufficient conditions are provided for identifying the class of nonlinear systems for which state observers with linear error dynamics can be obtained

an algorithm for orbital feedback linearization of single input control affine systems feedback linearization is an effective design and analysis tool used in the study of nonlinear control systems however cases arise where linearizability conditions cannot be met for systems that are not linearizable by classical techniques orbital feedback linearization or feedback linearization by state dependent time scaling has been proposed to relax these conditions unfortunately approaches proposed to date have led to conditions that tend to be more difficult to check then conditions for state feedback linearization in this paper necessary and sufficient conditions for orbital feedback linearizability is presented for a class of single input nonlinear systems the conditions are simple and can be checked directly from the data of the problem using an exterior calculus approach a simple algorithm is developed to compute state dependent time scaling that yield state feedback linearizable systems it is shown that orbital feedback linearizability generalizes the concept state feedback linearizability to deal with locally weakly accessible control systems

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible

generating functions fibonacci numbers and rational knots abstract we describe rational knots with any of the possible combinations of the properties a chirality non positivity non fiberedness and unknotting number one or higher and determine exactly their number for a given number of crossings in terms of their generating functions we show in particular how fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots then we show how to enumerate rational knots of given crossing number depending on genus and or signature this allows to determine the asymptotical average value of these invariants among rational knots we give also an application to the enumeration of lens spaces