on the noetherianity of some associative finitely presented algebras abstract we consider finitely generated associative algebras over a fixed field k of arbitrary characteristic for such an algebra a we impose some structural restrictions we call a strictly ordered we are interested in the implication of strict order on a for its noetherian properties in particular we prove that if a is a graded standard finitely presented strictly ordered algebra then a is left noetherian if and only if it is almost commutative in this case a has polynomial growth

matched pairs approach to set theoretic solutions of the yang baxter equation we study set theoretic solutions x r of the yang baxter equations on a set x in terms of the induced left and right actions of x on itself we give a characterisation of involutive square free solutions in terms of cyclicity conditions we characterise general solutions in terms of abstract matched pair properties of the associated monoid s x r and we show that r extends as a solution rs on s x r as a set finally we study extensions of solutions both directly and in terms of matched pairs of their associated monoids we also prove several general results about matched pairs of monoids s of the required type including iterated products s s s equivalent to rs a solution and extensions s t rs t examples include a general double construction s s rs s and some concrete extensions their actions and graphs based on small sets

right l groups geometric garside groups and solutions of the quantum yang baxter equation abstract right lattice ordered groups are introduced and studied as a general framework for garside groups and related groups without a garside element every right l group g has a two sided partially ordered subgroup n g which generalizes the quasi centre of an artin tits group the group n g splits into copies of z if g is noetherian the positive cone of a right l group is described as a structure that is known from algebraic logic a pair of left and right self similar hoops for noetherian right l groups g modularity of the lattice structure is characterized in terms of an operation on the set x g of atoms it is proved that modular garside groups are equivalent to finite projective spaces with a non degenerate labelling a concept of duality for x g is introduced and applied in the distributive case this gives a one to one correspondence between noetherian right l groups with duality and non degenerate unitary set theoretic solutions of the quantum yang baxter equation the description of garside groups via garside germs is extended to right l groups which yields a one sided enhancement and a new proof of dvurecenskij s non commutative extension of mundici s correspondence between abelian l groups and mv algebras

analysis of the asymptotic properties of the moesp type of subspace algorithms the moesp type of subspace algorithms are used for the identification of linear discrete time finite dimensional state space systems they are based on the geometric structure of covariance matrices and exploit the properties of the state vector extensively in this paper the asymptotic properties of the algorithms are examined the main results include consistency and asymptotic normality for the estimates of the system matrices under suitable assumptions on the noise sequence the input process and the underlying true system

the asymptotic variance of subspace estimates abstract we give new simple general expressions for the asymptotic covariance of the estimated system parameters a b c d in subspace identification the formulas can be applied to a whole class of subspace methods including n4sid moesp cva etc the asymptotic expressions highlight how the conditioning of the estimation problem influences the accuracy of the estimates

multipermutation solutions of the yang baxter equation set theoretic solutions of the yang baxter equation form a meeting ground of mathematical physics algebra and combinatorics such a solution consists of a set x and a function r x x x x which satisfies the braid relation

representations of jointly stationary stochastic feedback processes abstract stable constant linear closed loop systems relating an input vector u to an output vector u and vice versa produce a jointly stationary y u procoss on the other hand it is often natural to split up a stationary vector random process z into component vectors yand u and to examine the closed loop relations between y and u this paper presents a number of new results on the spectral factorization and the closed loop representation of a jointly stationary vector y u proccss conditions are derived on the closed loop models for the joint process model to bo oe minimal degree stable and minimum phase relations between different joint process models producing the same spectrum yu z are established

a covering theory for non involutive set theoretic solutions to the yang baxter equation abstract an algebraic theory of non involutive set theoretic solutions to the yang baxter equation by means of a non commutative version of braces is established the structure group and the permutation group of a solution is put into a context of discrete lie theory this leads to a natural concept of covering which shares the good properties of coverings in geometry and topology in the special case of quandle coverings in knot theory known anomalies are removed

a necessary and sufficient condition for stability of a perturbed system the problem of determining the stability of a feedback system in the presence of perturbation is considered a necessary and sufficient condition under which the perturbed system remains stable is obtained this condition is very general in particular it does not require the perturbed plant to have the same number of poles as the nominal plant

structure smith macmillan form and coprime mfds of a rational matrix inside a region p ω abstract the structure of the smith macmillan form of a rational matrix t s inside a region p where is asymmetric with respect to the real axis subset of the finite complex plane c is determined algorithmic procedures based on elementary row and column operations over the euclidean ring r p s consisting of all rational functions with no poles in p are given coprimeness in p of a pair of rational matrices is studied in detail these results lead to constructive procedures for determining the coprime in p matrix fraction descriptions of t a

brief bezout identity related to reduced order observer based controllers for singular systems a design method of causal reduced order observer based controllers for singular systems is presented a new state space representation of the bezout identity for singular systems is developed a parameterization of all causal reduced order controllers is also given the general results given here can be extended to various cases including minimal and full order observer based controllers

a decomposition theorem for square free unitary solutions of the quantum yang baxter equation abstract it is known that every skew polynomial ring with generating set x and binomial relations in the sense of gateva ivanova trans amer math soc 343 1994 203 is an artin schelter regular domain of global dimension x moreover every such ring gives rise to a non degenerate unitary set theoretical solution r x 2 x 2 of the quantum yang baxter equation which fixes the diagonal of x 2 gateva ivanova s conjecture talk at the international algebra conference miskolc hungary 1996 states that conversely every such solution r comes from a skew polynomial ring with binomial relations an equivalent conjecture duke math j 100 1999 169 says that the underlying set x is r decomposable we prove these conjectures and construct an indecomposable solution r with x which shows that an extension to infinite x is false

feedback system design the tracking and disturbance rejection problems the problem of designing a compensator for a specified plant which simultaneously stabilizes the resultant feedback system and causes it to track a prescribed family of inputs and or reject prescribed disturbances is considered a set of linear design equations in the space of stable systems is formulated in a general linear systems setting and an explicit parameterization of the resultant solution space is obtained for a class of generalized multivariate systems the theory is illustrated with several single and multivariate examples

on consistency of subspace methods for system identification subspace methods for identification of linear time invariant dynamical systems typically consist of two main steps first a so called subspace estimate is constructed this first step usually consists of estimating the range space of the extended observability matrix secondly an estimate of system parametersis obtained based on the subspace estimate in this paper the consistency of a large class of methods for estimating the extended observability matrix is analyzed persistence of excitation conditions on the input signal are given which guarantee consistent estimates for systems with only measurement noise for systems with process noise it is shown that a persistence of excitation condition on the input is not sufficient more precisely an example for which the subspace methods fail to give a consistent estimate of the transfer function is given this failure occurs even if the input is persistently exciting of any order it is also shown that this problem can be eliminated if stronger conditions on the input signal are imposed

necessary and sufficient condition for stability robustness under known additive perturbation a necessary and sufficient condition is derived for the stability robustness of a unity feedback closed loop system involving a strictly proper plant p0 and a stabilizing compensator c0 under the assumption that the plant p0 is perturbed to p0 p0 where p0 is a known strictly proper rational matrix

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids abstract we consider algebras over a field k with generators x 1 x 2 x n subject to n 2 quadratic relations of the form x i x j x k x l with i j k l and moreover every monomial x i x j appears at most once in one of the defining relations if these relations are non degenerate then it is shown that the algebra is left and right noetherian satisfies a polynomial identity and has gelfand kirillov dimension at most n in case the defining relations are square free this was already established by gateva ivanova jespers and okninski to prove these results we investigate the structure of the underlying monoid s defined by the same presentation it is called a quadratic monoid we show that there is a strong link with the divisibility monoids and monoids of i type also referred to as yb monoids monoids of i type are examples of non degenerate quadratic monoids they are a monoid interpretation of non degenerate involutive set theoretic solutions of the yang baxter equation and have been studied quite intensively in recent years by gateva ivanova and van den bergh etingof schedler and soloviev jespers and okninski divisibility monoids have been introduced by kuske these are cancellative monoids that include the class of monoids of i type we show that they have a presentation with at most n 2 relations and if they have precisely n 2 defining relations then they are monoids of i type

analysis of state space system identification methods based on instrumental variables and subspace fitting subspace based state space system identification 4sid methods have recently been proposed as an alternative to more traditional techniques for multivariable system identification the advantages are that the user has simple and few design variables and that the methods have robust numerical properties and relatively low computational complexities though subspace techniques have been demonstrated to perform well in a number of cases the performance of these methods is neither fully understood nor analyzed our principal objective is to undertake a statistical investigation of subspace based system identification techniques the studied methods consist of two steps the subspace spanned by the extended observability matrix is first estimated the asymptotic properties of this subspace estimate are derived herein in the second step the structure of the extended observability matrix is used to find a system model estimate two possible methods are considered the simplest one only uses a certain shift invariance property while in the other method a parametric representation of the null space of the observability matrix is exploited explicit expressions for the asymptotic estimation error variances of the corresponding pole estimates are given

noetherian properties of skew polynomial rings with binomial relations in this work we study standard finitely presented associative algebras over a fixed field k a restricted class of skew polynomial rings with quadratic relations considered in an earlier work of m artin and w schelter will be studied we call them binomial skew polynomial algebras we establish necessary and sufficient conditions for such an algebra to be a noetherian domain

an introduction to commutative and noncommutative gro bner bases abstract in 1965 buchberger introduced the notion of grobner bases for a polynomial ideal and an algorithm buchberger algorithm for their computation since the end of the seventies grobner bases have been an essential tool in the development of computational techniques for the symbolic solution of polynomial systems of equations and in the development of effective methods in algebraic geometry and commutative algebra moreover grobner bases have been also generalized to free noncommutative algebra and to various noncommutative algebras of interest in differential algebra e g weyl algebras enveloping algebras of lie algebras the aim of this paper is to give an introduction as elementary as i was able to make it to both commutative and noncommutative algebras grobner bases are in a sense a finite model of an infinite linear gauss reduced basis of an ideal viewed as a vector space and buchberger algorithm is the corresponding generalization of the gaussian elimination algorithm moreover the paper contains a survey of some applications of buchberger theory to noncommutative algebras together with these results surveyed this paper contains some minor new points e g the useless pair criteria in the noncommutative case and the final result on the existence and computability of grobner bases for two sided ideals in any finitely presented algebra

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

on and off line identification of linear state space models abstract a geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time invariant systems using possibly noisy input output i o measurements only as opposed to other mostly stochastic identification schemes no variance covariance information whatever is involved and only a limited number of i o data are required for the determination of the system matrices hence the algorithm can be best described and understood in the matrix formalism and consists of the following two steps first a state vector sequence is realized as the intersection of the row spaces of two block hankel matrices constructed with i o data then the system matrices are obtained at once from the least squares solution of a set of linear equations when dealing with noisy data this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition also the algorithm is easily applied to slowly time

asymptotic variance of closed loop subspace identification methods in this paper the asymptotic properties of a version of the innovation estimation algorithm by qin and ljung as well as of a version of the whitening filter based algorithm introduced by jansson are studied expressions for the asymptotic error as the sum of a bias term plus a variance term are given the analysis is performed under rather mild assumptions on the spectrum of the joint input output process however in order to avoid unnecessary complications the asymptotic variance formulas are computed explicitly only for finite memory systems i e of the arx type this assumption could be removed at the price of some technical complications the simulation results confirm that when the past horizon is large enough as compared to the predictor dynamics the asymptotic expressions provide a good approximation of the asymptotic variance also for armax systems

technical communique asymptotic variance expressions for closed loop identification asymptotic variance expressions are analyzed for models that are identified on the basis of closed loop data the considered methods comprise the classical direct method as well as the more recently developed indirect methods employing coprime factorized models dual youla kucera parametrizations and the two stage approach the variance expressions are compared with the open loop situation and evaluated in terms of their relevance for subsequent model based control design

closed loop identification revisited identification of systems operating in closed loop has long been of prime interest in industrial applications the problem offers many possibilities and also some fallacies and a wide variety of approaches have been suggested many quite recently the purpose of the current contribution is to place most of these approaches in a coherent framework thereby showing their connections and display similarities and differences in the asymptotic properties of the resulting estimates the common framework is created by the basic prediction error method and it is shown that most of the common methods correspond to different parameterizations of the dynamics and noise models the so called indirect methods e g are indeed direct methods employing noise models that contain the regulator the asymptotic properties of the estimates then follow from the general theory and take different forms as they are translated to the particular parameterizations we also study a new projection approach to closed loop identification with the advantage of allowing approximation of the open loop dynamics in a given and user chosen frequency domain norm even in the case of an unknown nonlinear regulator

subspace model identification part 1 the output error state space model identification class of algorithms in this paper we present two novel algorithms to realize a finite dimensional linear time invariant state space model from input output data the algorithms have a number of common features they are classified as one of the subspace model identification schemes in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input output data this structure is then exploited in the calculation of a realization another common feature is their algorithmic organization an rq factorization followed by a singular value decomposition and the solution of an overdetermined set or sets of equations the schemes assume that the underlying system has an output error structure and that a measurable input sequence is available the latter characteristic indicates that both schemes are versions of the mimo output error state space model identification moesp approach the first algorithm is denoted in particular as the elementary moesp scheme

set theoretic solutions of the yang baxter equation braces and symmetric groups abstract we involve simultaneously the theory of braided groups and the theory of braces to study set theoretic solutions of the yang baxter equation ybe we show the intimate relation between the notions of a symmetric group in the sense of takeuchi i e a braided involutive group and a left brace we find new results on symmetric groups of finite multipermutation level and the corresponding braces we introduce a new invariant of a symmetric group g r the derived chain of ideals of g which gives a precise information about the recursive process of retraction of g we prove that every symmetric group g r of finite multipermutation level m is a solvable group of solvable length m to each set theoretic solution x r of ybe we associate two invariant sequences of involutive braided groups i the sequence of its derived symmetric groups ii the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction of x r we find new criteria necessary and sufficient to claim that x r is a multipermutation solution