recursive identification using feedforward neural networks the paper is concerned with the identification of an unknown nonlinear dynamical system when only the inputs and outputs are accessible for measurement under certain assumptions it is shown that generically the system can be realized by a recursive input output model furthermore relying on the approximation properties of neural networks and the existence of effective training algorithms it is demonstrated how an effective identification model can be constructed simulation results are presented to complement the theoretical discussions

neural networks for functional approximation and system identification we construct generalized translation networks to uniformly approximate a class of nonlinear continuous functionals defined on l p 1 1 s for integer s 1 1 p 1 or c 1 1 s we obtain lower bounds on the possible order of approximation for such functionals in terms of any approximation process depending continuously upon a given number of parameters our networks almost achieve this order of approximation in terms of the number of parameters neurons involved in the network the training is simple and noniterative in particular we avoid any optimization such as that involved in the usual back propagation

identification of systems containing linear dynamic and static nonlinear elements identification of nonlinear systems which can be represented by combinations of linear dynamic and static nonlinear elements are considered previous results by the authors based on correlation analysis are combined to provide a unified treatment for this class of systems it is shown that systems composed of cascade feedforward feedback and multiplicative connections of linear dynamic and zero memory nonlinear elements can be identified in terms of the individual component subsystems from measurements of the system input and output only

minimal realization of nonlinear systems described by input output difference equations we consider a general class of nonlinear discrete time systems described by their input output i o relationship we first present the necessary and sufficient conditions for existence of a local observable state space realization of the system and an explicit algorithm for computing it in cases where the i o map and its resulting realization are nonminimal we formulate an algorithm for extracting its minimal realization whenever possible the developed theory and algorithms are illustrated by means of several examples

the determination of optimum structures for the state space representation of multivariate stochastic processes when identifying a model for a multivariate stationary stochastic process an important problem is that of determining the structure of the state variable model several overlapping parameterizations can usually be fitted to a given process and the question arises as to which structure leads to the most accurate parameter estimates the accuracy of parameter estimates is often measured by the determinant of the fisher information matrix we show that all admissible structures will give asymptotically the same value to this criterion for finite data some structures may still be better than others and two heuristic structure estimation methods are analyzed some simulation results are also presented

analytic models and ambiguity of context free languages we establish that several classical context free languages are inherently ambiguous by proving that their counting generating functions when considered as analytic functions exhibit some characteristic form of transcendental behaviour to that purpose we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages in particular the paper contains a general density theorem for unambiguous context free languages

polynomial time reductions from multivariate to bi and univariate integral polynomial factorization consider a polynomial f with an arbitrary but fixed number of variables and with integral coefficients we present an algorithm which reduces the problem of finding the irreducible factors of f in polynomial time in the total degree of f and the coefficient lengths of f to factoring a univariate integral polynomial together with a lenstra s h lenstra s and l lovasz polynomial time factorization algorithm for univariate integral polynomials math ann 261 1982 pp 515 534 this algorithm implies the following theorem factoring an integral polynomial with a fixed number of variables into irreducibles except for the constant factors can be accomplished in deterministic polynomial time in the total degree and the size of its coefficients our algorithm can be generalized to factoring multivariate polynomials with coefficients in algebraic number fields and finite fields in polynomial time we also present a different algorithm based on an effective version of a hilbert irreducibility theorem w

identification using feedforward networks this paper is concerned with the identification of an unknown nonlinear dynamic system when only the inputs and outputs are accessible for measurement specifically we investigate the use of feedforward neural networks as models for the input output behavior of such systems relying on the approximation capabilities of feedforward neural networks and under mild assumptions regarding the properties of the underlying nonlinear system it is shown that there exists a feedforward network that for almost all inputs an open and dense set will display the input output behavior of the system

dynamical realizations of finite volterra series in this paper realizations of finite volterra series are viewed as nonlinear analytic input output systems with state space described by an analytic manifold for a minimal realization guaranteed by h j sussmann the state space which is unique up to diffeomorphism is shown to have the homogeneous space structure of a nilmanifold the quotient of two nilpotent lie groups the structure of nilmanifolds as described by a malcev is used to show that for these systems the state space has a vector space structure as a consequence of this result it is shown that a minimal realization of a finite volterra series can be described as a cascade of linear subsystems with polynomial link maps in which the dimension o f each linear subsystem is independent of the realization considered

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

weak hopf algebras i integral theory and c structure we give an introduction to the theory of weak hopf algebras proposed as a coassociative alternative of weak quasi hopf algebras we follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras the emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of non commutative numbers in many respects a theory of integrals is developed in which we show how the algebraic properties of a such as the frobenius property or semisimplicity or innerness of the square of the antipode are related to the existence of non degenerate normalized or haar integrals in case of c weak hopf algebras we prove the existence of a unique haar measure h a and of a canonical grouplike element g a implementing the square of the antipode and factorizing into left and right elements g glg 1r gl al gr ar further discussion of the c case will be presented in part ii

three dimensional quantum groups from contractions of su 2 q contractions of lie algebras and of their representations are generalized to define new quantum groups an explicit and complete exposition is made for the one dimensional heisenberg h 1 q and the two dimensional euclidean quantum group e 2 q obtained by contracting su 2 q

quantumsu 2 ande 2 groups contraction procedure in 3 it was shown in the framework of deformed enveloping algebras that quantumsu 2 ande 2 groups are related by the contraction procedure we consider the same problem on thec level as a result we find a number of formulae coupling the comultiplications in quantumsu 2 ande 2 in particular we show that the comultiplications in both groups are implemented by partial isometries an unexpected feature of quantume 2 is discovered and the corresponding strange behavior of quantumsu 2 is described

operator equalities related to the quantum e 2 group the paper deals with normal operatorsr ands satisfying simple commutation relations that we encounter investigating the quantum deformation of thee 2 group we show thatr s admits a normal extension if and only ifr 1s satisfies a certain spectral condition a number of related formulae are derived in particular all the functionsf satisfying the character equationf r s f r f s are found

effective entropies and data compression abstract in this paper we introduce the formal definition of the concept of the c 1 c 2 effective entropy of a language where c 1 c 2 are complexity classes the c 1 c 2 effective entropy of a language l is used to measure how much a string x of length n in l can be compressed to a string y by a c 1 algorithm so that given y x can be recovered by a c 2 algorithm we also relate this concept to issues in data compression the main results are 1 the scj effective entropy of a 2o logn sparse regular language is equal its abosolute entropy i e it is optimal 2 the det sc j effective entropy of a 2o logn sparse 2o logn ambiguous linear context free language is up to a constant factor equal its absolute entropy det denotes the class of problems which are nc 1 reducible to computing the determinants of integer matrices and 3 the nc 2 p effective entropy of a 2o logn sparse 2o logn ambiguous context free language is up to a constant factor equal its absolute entropy

hopf galois systems abstract we introduce the concept of hopf galois system a reformulation of the notion of galois extension of the base field for a hopf algebra the main feature of our definition is a generalization of the antipode of an ordinary hopf algebra we present several examples which indicate that although our axiomatic is slightly more complicated than the classical one it is also more natural and easier to handle with the main application of hopf galois systems is the construction of monoidal equivalences between comodule categories

deformation of a kac algebra by an abelian subgroup some examples of quantum groups in literature arise as deformations of a locally compact group by a dual 2 cocycle we make this construction in the framework of kac algebras we show that these deformations are still kac algebras using this construction we give new quantizations of the heisenberg group from this point of view we analyse the dimension 8 non trivial example of kac and paljutkin and give a new example of non trivial dimension 12 semi simple hopf algebras a dimension 12 kac algebra

computing a context free grammar generating series the parallel complexity of computing context free grammar generating series is investigated it is known that this problem is in div but in terms of n rather than n where n is the index of the desired coefficient and is the grammar size a new method is presented which is in div in terms of 22o n evidence is provided that any direct application of elimination theory to this problem leads to a space and time resource factor that is nearly exponential in grammar size

logarithmic depth circuits for algebraic functions this paper describes circuits for computation of a large class of algebraic functions on polynomials power series and integers for which it has been a long standing open problem to compute in depth less than omega log n 2 algebraic circuits assume unit cost for elemental addition and multiplication this paper describes o log n depth algebraic circuits which given as input the coefficients of n degree polynomials over an appropriate ring compute the product of n o 1 polynomials the symmetric functions as well as division and interpolation of real polynomials also described are o log n depth algebraic circuits which are given as input the first n coefficients of a power series over an appropriate ring compute the product of n o 1 power series as well as division reciprocal and reversion of real power series furthermore this paper describes boolean circuits of depth o log n log log n which given n bit binary numbers compute the product of n numbers and integ

an explicit description of the fundamental unitary for su 2 q we give a concrete description of an isometryv from l2 to l2 whose existence has recently been discovered by woronowicz 11 the isometryv gives the comultiplication on the c algebraa of the quantum group su 2 q through the formula x v x 1 v x a where 1 is the identity operator on l2 the matrix entries ofv are described in terms of littleq jacobi polynomials usingv we give a concrete description of a unitary operatorv onh h such that x v x 1 v whereh l2 and a l h is the gns representation associated with the haar state ona the operatorv satisfies the pentagonal identity of baaj and skandalis 1

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

existence and uniqueness of realizations of nonlinear systems necessary and sufficient conditions for the existence of either analytic or smooth symmetric realizations of nonlinear input output maps are given it is also shown that two minimal realizations are diffeomorphic

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

on cocycle twisting of compact quantum groups abstract we provide a class of examples of compact quantum groups and unitary 2 cocycles on them such that the twisted quantum groups are non compact but still locally compact quantum groups in the sense of kustermans and vaes this also gives examples of cocycle twists where the underlying c algebra of the quantum group changes

on a morita equivalence between the duals of quantum su 2 and quantum e 2 abstract let su q 2 and e q 2 be woronowicz s q deformations of respectively the compact lie group su 2 and the non trivial double cover of the lie group e 2 of euclidean transformations of the plane we prove that in some sense their duals are morita equivalent locally compact quantum groups in more concrete terms we prove that the von neumann algebraic quantum groups l su q 2 and l e q 2 are unitary cocycle deformations of each other

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

deformation of c algebras by cocycles on locally compact quantum groups given a c algebra a with a left action of a locally compact quantum group g on it and a unitary 2 cocycle on g we define a deformation a of a the construction behaves well under certain additional technical assumptions on the most important of which is regularity meaning that c0 g g is isomorphic to the algebra of compact operators on some hilbert space in particular then a is stably isomorphic to the iterated twisted crossed product g op g a also in good situations the c algebra a carries a left action of the deformed quantum group g and we have an isomorphism g a g a when g is a genuine locally compact group we show that the action of g on c0 g cr g is always integrable stronger assumptions of properness and saturation of the action imply regularity as an example we make a preliminary analysis of the cocycles on the duals of some solvable lie groups recently constructed by bieliavsky et al and discuss the relation of our construction to that of bieliavsky and gayral

generic observability of differentiable systems a dynamical system consists of a smooth vectorfield defined on a differentiable manifold and a smooth mapping from the manifold to the real numbers the vectorfield represents the dynamics of a physical system the mapping stands for a measuring device by which experimental information on the dynamics is made available the information itself is modeled as a sampled version of the image of the state trajectory under the smooth mapping in this paper the observability of this set up is discussed from the viewpoint of genericity first the observability property is expressed in terms of transversality conditions then the theory of transversal intersection is called upon to yield the desired results it is shown that almost any measuring device will combine with a given physical system to form an observable dynamical system if 2n 1 samples are taken and not fewer where n is the dimension of the manifold dually it is shown that almost any physical system will combine with a given measuring device

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem