the prime spectrum of algebras of quadratic growth abstract we study prime algebras of quadratic growth our first result is that if a is a prime monomial algebra of quadratic growth then a has finitely many prime ideals p such that a p has gk dimension one this shows that prime monomial algebras of quadratic growth have bounded matrix images we next show that a prime graded algebra of quadratic growth has the property that the intersection of the non zero prime ideals p such that a p has gk dimension 2 is non zero provided there is at least one such ideal from this we conclude that a prime monomial algebra of quadratic growth is either primitive or has non zero locally nilpotent jacobson radical finally we show that there exists a prime monomial algebra a of gk dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that a p has gk dimension 1

semiprime graded algebras of dimension two semiprime noetherian connected graded k algebras r of quadratic growth are described in terms of geometric data a typical example of such a ring is obtained as follows let y be a projective variety of dimension at most one over the base field k and let e be an oy order in a finite dimensional semisimple algebra a over k k y then for any automorphism of a that restricts to an automorphism of y and any ample invertible e bimodule b van den bergh constructs a noetherian twisted homogeneous coordinate ring b h0 y b b n 1 we show that r is noetherian if and only if some veronese ring r m of r has the form k i where i is a left ideal of such a ring b and where i b at each point p y at which has finite order this allows one to give detailed information about the structure of r and its modules

stability analysis of discrete time lur e systems a class of lyapunov functions is proposed for discrete time linear systems interconnected with a cone bounded nonlinearity using these functions we propose sufficient conditions for the global stability analysis in terms of linear matrix inequalities lmi only taking the bounded sector condition into account unlike frameworks based on the lur e type function the additional assumptions about the derivative or discrete variation of the nonlinearity are not necessary hence a wider range of cone bounded nonlinearities can be covered we also show that there is a link between global stability lmi conditions based on this new lyapunov function and a transfer function of an auxiliary system being strictly positive real in addition the novel function is considered in the local stability analysis problem of discrete time lur e systems subject to a saturating feedback a convex optimization problem based on sufficient lmi conditions is formulated to maximize an estimate of the basin of attraction another specificity of this new lyapunov function is the fact that the estimate is composed of disconnected sets numerical examples reveal the effectiveness of this new lyapunov function in providing a less conservative estimate with respect to the quadratic function

brief paper a new concept of invariance for saturated systems in this paper the notions of sns invariance and sns domain of attraction are introduced the sns domain of attraction serves as an estimation of the domain of attraction of a saturated system it is shown that in the case of single input saturated systems any contractive set is contained in the sns domain of attraction another important characteristic of the sns domain of attraction is that it contains any estimation obtained by means of a linear difference inclusion of the saturated system a simple algorithm that converges to the sns domain of attraction is presented an illustrative example is given

conjugate lyapunov functions for saturated linear systems based on a recent duality theory for linear differential inclusions ldis the condition for stability of an ldi in terms of one lyapunov function can be easily derived from that in terms of its conjugate function this paper uses a particular pair of conjugate functions the convex hull of quadratics and the maximum of quadratics for the purpose of estimating the domain of attraction for systems with saturation nonlinearities to this end the nonlinear system is locally transformed into a parametertized ldi system with an effective approach which enables optimization on the parameter of the ldi along with the optimization of the lyapunov functions the optimization problems are derived for both the convex hull and the max functions and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids a numerical example demonstrates the effectiveness of this paper s methods

an nth step set invariance approach to the analysis and design of discrete time linear systems subject to actuator saturation abstract for a discrete time linear system subject to actuator saturation an analysis and design approach based on n th step set invariance is proposed an n th step invariant set is a set from which all trajectories will enter a contractively invariant set in n steps and thus can be used as an estimate of the domain of attraction such an estimate is less conservative than an estimate obtained on the basis of contractive invariance of ellipsoids which excludes initial conditions from which the trajectories overshoot outside the ellipsoid before converging to the origin in comparison with the existing approach based on conventional set invariance the proposed approach results in a larger domain of attraction and better performance both in the analysis of the closed loop system under a given feedback gain and in the design of feedback gain in comparison with the saturated and nonsaturated sns invariant set approach which is an analysis approach the proposed approach can be used for control design even in the presence of plant uncertainties

an analysis and design method for discrete time linear systems under nested saturation this paper considers a discrete time linear system under nested saturation nested saturation arises for example in systems with actuators subject to both magnitude and rate saturation a condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant moreover this condition is shown to be equivalent to linear matrix inequalities lmis in the actual and auxiliary feedback gains as a result the estimation of the domain of attraction for a given set of feedback gains is then formulated as an optimization problem with lmi constraints by viewing the feedback gains as extra free parameters the optimization problem can be used for controller design

on the effect of noise on the moore penrose generalized inverse associative memory monte carlo simulations of the continuous moore penrose generalized inverse associative memory kohonen l have shown that the noise to signal ratio is improved on recall in the autoassociative case as long as the number of vector pairs stored is less than the number of components per vector in the heteroassociative case however the noise to signal ratio may actually be greatly increased upon recall particularly as the number of vector pairs stored approaches the number of components per vector the increase in output noise to signal ratio in the heteroassociative case is found to be due to the fact that the inverse of the product of the key vector matrix with its transpose may increase without bound in spite of the fact that the key vectors are linearly independent

golod shafarevich algebras free subalgebras and noetherian images abstract it is shown that golod shafarevich algebras of a reduced number of defining relations contain noncommutative free subalgebras in two generators and that these algebras can be homomorphically mapped onto prime noetherian algebras with linear growth it is also shown that golod shafarevich algebras of a reduced number of relations cannot be nil

a quantitative comparison of the performance of three discrete distributed associative memory models monte carlo methods have been used to study the recall accuracy of three discrete distributed associative memory models a discrete correlation matrix model dcm the bayes discriminant rule model developed by murakami and aibara ma and a discrete version of the generalized inverse model of kohonen dgi the key input and data output vectors have nc components restricted to the values 1 and 1 the elements of the nc nc recall matrix may take on any floating point value auto associative key and data vectors identical and hetero associative operation have been examined with both noiseless perfect and noisy key vectors simulations have been performed over a range of nc from 20 to 100 and of nv the number of vector pairs stored from 1 to 50 the recall accuracy defined as the probability that a component of the output vector is correct depends strongly on the ratio nv nc the dgi model is the only one able to provide perfect hetero associative recall with perfect input for nv nc but is extremely sensitive to input noise if the probability that a component of the input vector is correct is 0 9 the three models will provide perfect output only if nv is less than about 0 15 nc for more noisy inputs the ma model provides the highest probability that a component of the output will be correct

saturated feedback stabilization of discrete time descriptor bilinear systems this note addresses global asymptotic stabilization of a class of discrete time descriptor bilinear systems via a saturated state feedback controller based on lasalle invariance principle and the implicit function theorem sufficient conditions are presented to guarantee the uniqueness and existence of solution and the global asymptotic stability of the resulting closed loop systems simultaneously it is shown that the condition is independent of the state transformation for the given descriptor systems furthermore the effectiveness of the proposed design method is illustrated by a numerical example

growth of monomial algebras simple rings and free subalgebras abstract we construct finitely generated simple algebras with prescribed growth types which can be arbitrarily taken from a large variety of super polynomial growth types this partially answers a question raised in 9 question 5 1 our construction goes through a construction of finitely generated just infinite primitive monomial algebras with prescribed growth type from which we construct uniformly recurrent infinite words with subword complexity having the same growth type we also discuss the connection between entropy of algebras and their homomorphic images as well as the degrees of their generators of free subalgebras

a class of algebras similar to the enveloping algebra of sl 2 fix f e c x define r c a b h subject to the relations ha ah a hb bh b ab ba f h we study these algebras for different f and in particular show how they are similar to and different from u sl 2 the enveloping algebra of sl 2 c there is a notion of highest weight modules and a category a for such r for each n 0 if f x x 1 n l xn 1 then r has precisely n simple modules in each finite dimension and every finite dimensional r module is semisimple

quadratic stability for hybrid systems with nested saturations the problems of characterizing quadratic stability and computing an estimation of the domain of attraction for saturated hybrid systems are addressed hybrid systems presenting saturations and nested saturations on signals involved in both the continuous time and the discrete time dynamics are considered geometrical characterizations of local and global quadratic stability are provided computation oriented conditions for quadratic stability are given in form of convex constraints

computing regions of attraction with polytopes planar case abstract an algorithm that estimates the region of attraction of the origin using a convex polytope is developed for piecewise linear dynamical systems this is equivalent to the problem of computing a lyapunov function the r 2 case is studied here for simplicity

primitive algebras with arbitrary gelfand kirillov dimension abstract we construct for every real 2 a primitive affine algebra with gelfand kirillov dimension unlike earlier constructions there are no assumptions on the base field in particular this is the first construction over or l given a recursive sequence v n of elements in a free monoid we investigate the quotient of the free associative algebra by the ideal generated by all nonsubwords in v n we bound the dimension of the resulting algebra in terms of the growth of v n in particular if n is less than doubly exponential then the dimension is 2 this also answers affirmatively a conjecture of salwa 1997 comm algebra 25 3965 3972

simple associative conformal algebras of linear growth abstract we describe simple finitely generated associative conformal algebras of gel fand kirillov dimension one

examples in finite gel fand kirillov dimension for any field f we give an example of a prime finitely generated f algebra of gk dimension 2 for which gk dimension is not finitely partitive for each m 0 1 2 3 we give an example

saturated finite interval iterative learning for tracking of dynamic systems with hnn structural output this brief investigates the interval iterative learning problem for dynamic systems with hierarchical neural network hnn structural output the first objective is to design the output of a dynamic system with hnn structure a sufficient condition is obtained to achieve the interval tracking in a finite interval by applying iterative learning control ilc then the saturated ilc is considered into the discussed system and a less conservative criterion is obtained to achieve the tracking in a finite interval using a network structure decomposition technique finally simulation results are given to illustrate the usefulness of the developed criteria

exact associative neural memory dynamics utilizing boolean matrices the exact dynamics of shallow loaded associative neural memories are generated and characterized the boolean matrix analysis approach is employed for the efficient generation of all possible state transition trajectories for parallel updated binary state dynamic associative memories dams general expressions for the size of the basin of attraction of fundamental and oscillatory memories and the number of oscillatory and stable states are derived for discrete synchronous hopfield dams loaded with one two or three even dimensionality bipolar memory vectors having the same mutual hamming distances between them spurious memories are shown to occur only if the number of stored patterns exceeds two in an even dimensionality hopfield memory the effects of odd versus even dimensionality memory vectors on dam dynamics and the effects of memory pattern encoding on dam performance are tested an extension of the boolean matrix dynamics characterization technique to other more complex dams is presented

noetherian connected graded algebras of global dimension 3 abstract let a be a noetherian connected graded algebra of global dimension 3 we show that a is regular in the sense of artin and schelter if one of the following conditions holds 1 a is generated by two elements 2 the graded simple module has a standard resolution 3 the degrees of minimal relations are the same this includes the quadratic case some general properties of a are studied without assuming regularity

on the codimension sequence of g simple algebras abstract let f be an algebraically closed field of characteristic zero and let g be a finite group in this paper we will show that the asymptotics of c n g a the g graded codimension sequence of a finite dimensional g simple f algebra a has the form n 1 dim f a e 2 dim f a n as conjectured by e aljadeff d haile and m natapov where is some positive real number and a e denotes the identity component of a in the special case where a is the algebra of m m matrices with an arbitrary elementary g grading we succeeded in calculating the constant explicitly

associative conformal algebras with finite faithful representation abstract the classification of irreducible subalgebras of the associative conformal algebra cend n is presented in this paper the structure theory of associative conformal algebras with finite faithful representation is developed

free subalgebras of graded algebras let k be a field and let a n 1an be a positively graded k algebra we recall that a is graded nilpotent if for every d 1 the subalgebra of a generated by elements of degree d is nilpotent we give a method of producing grading nilpotent algebras and use this to prove that over any base field k there exists a finitely generated graded nilpotent algebra that contains a free k subalgebra on two generators

on associative conformal algebras of linear growth ii abstract we classify unital associative conformal algebras of linear growth and provide new examples of such

stability and performance for saturated systems via quadratic and nonquadratic lyapunov functions in this paper we develop a systematic lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration the only assumptions we make about the system are well posedness of the algebraic loop and local stability problems to be considered include the estimation of the domain of attraction the reachable set under a class of bounded energy disturbances and the nonlinear l2 gain the regional analysis is established through an effective treatment of the algebraic loop and the saturation deadzone function this treatment yields two forms of differential inclusions a polytopic differential inclusion pdi and a norm bounded differential inclusion ndi that contain the original system adjustable parameters are incorporated into the differential inclusions to reflect the regional property the main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain lyapunov function where the pdi or the ndi is valid with quadratic lyapunov functions conditions for stability and performances are derived as linear matrix inequalities lmis to obtain less conservative conditions we use a pair of conjugate non quadratic lyapunov functions the convex hull quadratic function and the max quadratic function these functions yield bilinear matrix inequalities bmis as conditions for stability and guaranteed performance level the bmi conditions cover the corresponding lmi conditions as special cases hence the bmi results are guaranteed to be as good as the lmi results in most examples the bmi results are significantly better than the lmi results

brief paper squaring the circle an algorithm for generating polyhedral invariant sets from ellipsoidal ones this paper presents a new geometrical approach to the computation of polyhedral robustly positively invariant pi sets for general possibly discontinuous nonlinear discrete time systems possibly affected by disturbances given a b contractive ellipsoidal set e the key idea is to construct a polyhedral set that lies between the ellipsoidal sets be and e a proof that the resulting polyhedral set is contractive and thus pi is given and a new algorithm is developed to construct the desired polyhedral set the problem of computing polyhedral invariant sets is formulated as a number of quadratic programming qp problems the number of qp problems is guaranteed to be finite and therefore the algorithm has finite termination an important application of the proposed algorithm is the computation of polyhedral terminal constraint sets for model predictive control based on quadratic costs

robust filtering for discrete time systems with saturation and its application to transmultiplexers this paper considers the problem of robust filtering for discrete time linear systems subject to saturation a generalized dynamic filter architecture is proposed and a filter design method is developed our approach incorporates the conventional linear h sub 2 and h sub spl infin filtering as well as a regional l sub 2 gain filtering feature developed specially for the saturation nonlinearity and is applicable to the digital transmultiplexer systems for the purpose of separating filterbank design it turns out that our filter design can be carried out by solving a constrained optimization problem with linear matrix inequality lmi constraints simulations show that the resultant separating filters possess satisfactory reconstruction performance while working in the linear range and less degraded reconstruction performance in the presence of saturation

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

analysis and design for delta operator systems with actuator saturation this paper studies the problem of estimating the domain of attraction for delta operator systems under a pre designed saturated linear feedback ellipsoids are used as the estimates of domain of attraction for delta operator systems a condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractive invariant moreover some necessary and sufficient conditions are given for single input delta operator systems the results are also further extended to determine the invariant sets for delta operator systems with persistent disturbances linear matrix inequalities based on the methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements some numerical examples are given to illustrate the effectiveness of the developed techniques

stochastic finite time boundedness on switching dynamics markovian jump linear systems with saturated and stochastic nonlinearities in this paper problems of finite time boundedness are investigated for a class of discrete time switching dynamics markovian jump linear systems with saturated and stochastic nonlinearities the time varying transition probabilities are described by a piecewise constant matrix subject to a high level average dwell time adt switching sensor and actuator saturations are characterized by a vector valued decomposition method and the stochastic nonlinearities are approximated by a statistical method in general not all trajectories originating from the admissible initial states could be stabilized in the mean square sense or sufficient conditions are too restrictive to yield feasible solutions therefore the purpose of studying the problems addressed here is to design an output feedback controller via the adt approach such that the resulting closed loop systems are stochastically finite time bounded and have a guaranteed disturbance attenuation capability simulation results demonstrate the potential and effectiveness of the theoretical results

robust control of multi input periodic discrete time systems with saturating actuators this paper proposes the use of a time varying sliding surface for robust transient shaping of periodic linear discrete time plants subject to saturating actuators in the presence of bounded matched uncertainties a constructive procedure is presented and ultimate boundedness of state trajectories is proved

brief paper stability and stabilization of discrete time periodic linear systems with actuator saturation this paper is concerned with the problems of stability and stabilization for discrete time periodic linear systems subject to input saturation both local results and global results are obtained for local stability and stabilization the so called periodic invariant set is used to estimate the domain of attraction the conditions for periodic invariance of an ellipsoid can be expressed as linear matrix inequalities lmis which can be used for both enlarging the domain of attraction with a given controller and synthesizing controllers the periodic enhancement technique is introduced to reduce the conservatism in the methods as a by product less conservative results for controller analysis and design for discrete time time invariant systems with input saturation are obtained for global stability by utilizing the special properties of the saturation function a saturation dependent periodic lyapunov function is constructed to derive sufficient conditions for guaranteeing the global stability of the system the corresponding conditions are expressed in the form of lmis and can be efficiently solved several numerical and practical examples are given to illustrate the theoretical results proposed in the paper

controller design for nonlinear quadratic markov jumping systems with input saturation this paper deals with the controller design problem of nonlinear quadratic markov jumping systems with input saturation both mode dependent and mode independent state feedback controllers are designed by using the concept of domain of attraction in mean square sense sufficient conditions for stochastic stabilisation for nonlinear quadratic systems are derived in terms of linear matrix inequalities certain existing results are improved simulation examples are presented to illustrate the effectiveness of the proposed technique

a dichotomy result for prime algebras of gelfand kirillov dimension two abstract let k be an uncountable field we show that a finitely generated prime goldie k algebra of quadratic growth is either primitive or satisfies a polynomial identity answering a question of small in the affirmative

brief stability analysis of discrete time systems with actuator saturation by a saturation dependent lyapunov function in this paper stability of discrete time linear systems subject to actuator saturation is analyzed using a saturation dependent lyapunov function this saturation dependent lyapunov function captures the real time information on the severity of actuator saturation and leads to less conservative estimate of the domain of attraction which is based on the solution of an lmi optimization problem numerical examples are presented to show the effectiveness of the proposed method

there are no graded domains with gk dimension strictly between 2 and 3 let k be a field and let r n n r n be a finitely generated graded k algebra which is a domain it is shown that r cannot have gelfand kirillov dimension strictly between 2 and 3

ultimate boundedness control for uncertain discrete time systems via set induced lyapunov functions in this note linear discrete time systems affected by both parameter and input uncertainties are considered the problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence is investigated it is shown that the problem has a solution if and only if there exists a certain lyapunov function which does not belong to a preassigned class of functions e g the quadratic ones but it is determined by the target set in which ultimate boundedness is desired one of the advantages of this approach is that we may handle systems with control constraints no matching assumptions are made for systems with linearly constrained uncertainties it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets the resulting compensator may be implemented as a linear variable structure control

construction of a distributed associative memory on the basis of bayes discriminant rule the purpose of this correspondence is to propose a new construction method of distributed associative memory which operates with discrete valued signals in this method memorized pairs of vectors cue vectors and data vectors are recorded in the form of a matrix w and a vector t from an input vector x the data vector is recalled by an operation u xw t where x is a cue vector or a noisy cue vector and u is a quantizing function the methods of memorization and recall are similar to the associatron however the proposed model can recall the data vectors optimally in bayesian sense even when noisy cue vectors are given as the input vectors

augmented lyapunov function approach to l2 gain analysis for discrete time systems with saturation nonlinearities abstract this paper presents a method for l 2 gain analysis of discrete time systems with saturation nonlinearity by using an augmented lyapunov functional by using sector and slope properties of the saturation nonlinearity the l 2 gain performance criterion is obtained via lmi formulation the proposed lyapunov function have augmented quadratic form with states as well as the sector bounded nonlinear function numerical examples show that the proposed method is less conservative than the existing ones

analysis and design for discrete time linear systems subject to actuator saturation we present a method to estimate the domain of attraction for a discrete time linear system under a saturated linear feedback a simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant moreover the condition can be expressed as linear matrix inequalities lmis in terms of all the varying parameters and hence can easily be used for controller synthesis the following surprising result is revealed for systems with single input suppose that an ellipsoid is made invariant with a linear feedback then it is invariant under the saturated linear feedback if and only if there exists a saturated nonlinear feedback which makes the ellipsoid invariant finally the set invariance condition is extended to determine invariant sets for systems with persistent disturbances lmi based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements

paper maximal lyapunov functions and domains of attraction for autonomous nonlinear systems in this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system these methods are based on the concept of a maximal lyapunov function which is introduced in this paper a partial differential equation characterizing a maximal lyapunov function is derived and the relationships of this equation as compared to zubov s partial differential equation are discussed an iterative procedure is given for solving the new partial differential equation this procedure yields lyapunov function candidates that are rational functions rather than polynomials the method is applied to four two dimensional examples and one three dimensional example and it is shown that the estimates obtained using this method are in many cases substantially better than those obtained using known methods

linear discrete time global and regional anti windup an lmi approach in this article we address linear anti windup design for linear discrete time control systems guaranteeing regional and global stability and performance the techniques that we develop are the discrete time counterpart of existing techniques for anti windup augmentation which lead to convex constructions by way of linear matrix inequalities lmis when adopting static and plant order anti windup augmentation interesting system theoretic interpretations of the performance bounds for the non linear closed loop can also be given we show here that parallel results apply to the discrete time case we derive the corresponding conditions and prove their effectiveness by adapting the continuous time approaches to the discrete time case

some noncommutative minimal surfaces abstract in the ongoing programme to classify noncommutative projective surfaces connected graded noetherian domains of gelfand kirillov dimension three a natural question is to determine the minimal models within any birational class in this paper we show that the generic noncommutative projective plane corresponding to the three dimensional sklyanin algebra as well as noncommutative analogues of p 1 p 1 and the more general van den bergh quadrics satisfy very strong minimality conditions translated into an algebraic question where one is interested in a maximality condition we prove the following result theorem a let r be a sklyanin algebra or a van den bergh quadric that is infinite dimensional over its centre and let a r be any connected graded noetherian maximal order with the same graded quotient ring as r then up to taking veronese rings a is isomorphic to r let t be an elliptic algebra that is the coordinate ring of a noncommutative surface containing an elliptic curve then under an appropriate homological condition we prove that every connected graded noetherian overring of t is obtained by blowing down finitely many lines line modules of self intersection 1

the nilpotency of the radical in a finitely generated p i ring abstract let r x 1 x k be a p i ring satisfying a monic polynomial identity one of its coefficients is 1 where is a central noetherian subring it is proven that n r the nil radical of r is nilpotent as a corollary by taking f a field we settle affirmatively the open problem posed in c procesi rings with polynomial identities p 186 marcel dekker new york 1973 we prove the jacobson radical of a finitely generated p i algebra is nilpotent

second order counterexamples to the discrete time kalman conjecture the kalman conjecture is known to be true for third order continuous time systems we show that it is false in general for second order discrete time systems by construction of counterexamples with stable periodic solutions we discuss a class of second order discrete time systems for which it is true provided the nonlinearity is odd but false in general this has strong implications for the analysis of saturated systems

an upper bound for the length of a finite dimensional algebra abstract letfbe a field and letabe a finite dimensionalf algebra writed dimf a and letebe the largest degree of the minimal polynomial for anya a define the function f d e e 2d e 1 1 4 e 2 2 we prove that ifsis any finite generating set foraas anf algebra the words insof length less thanf d e spanaas anf vector space in the special case ofn by nmatrices this bound becomes f n 2 n n 2n 2 n 1 1 4 n 2 2 o n 3 2 this is a substantial improvement over previous bounds which have all beeno n2 we also prove that for particular setssof matrices the bound can be sharpened to one that is linear inn as an application of these results we reprove a theorem of small stafford and warfield about semiprime affinef algebras

on the kurosh problem for algebras of polynomial growth over a general field lenagan and smoktunowicz 2007 ls see also lenagan smoktunowicz and young in press lsy gave an example of a nil algebra of finite gelfand kirillov dimension their construction requires a countable base field however we show that for any field k and any monotonically increasing function f n which grows super polynomially but subexponentially there exists an infinite dimensional finitely generated nil k algebra whose growth is asymptotically bounded by f n this construction gives the first examples of nil algebras of subexponential growth over uncountable fields

h filtering with randomly occurring sensor saturations and missing measurements in chap 5 the h filtering problem is studied for a class of nonlinear systems with randomly occurring incomplete information where the considered incomplete information includes both the sensor saturations and the missing measurements a new phenomenon of sensor saturation namely randomly occurring sensor saturation ross is first put forward in order to better reflect the reality in a networked environment then a novel sensor model is established to account for both the ross and missing measurement in a unified representation based on this sensor model a regional h filter with a certain ellipsoid constraint is designed such that the filtering error dynamics is locally mean square asymptotically stable and the h norm requirement is satisfied

stabilization of switched linear systems subject to actuator saturation via invariant semiellipsoids this article presents new stabilization conditions for the state feedback control of switched linear systems subject to actuator saturation based on the semiellipsoidal approach and the multiple lyapunov functions method state feedback controllers and a state dependent switching law are designed the presented stabilization conditions are less conservative than those obtained by the ellipsoidal method and simpler than those established by the polyhedral method moreover the union of all positively invariant semiellipsoids is constructed to approximate the region of attraction each semiellipsoidal set is the intersection of a positively invariant ellipsoidal set and a constraint set constructive solutions of the presented conditions are obtained based on two different techniques the sum of squares technique and the nonlinear matrix inequality technique two illustrative examples are given to demonstrate the effectiveness of the main results

a magnitude and rate saturation model and its use in the solution of a static anti windup problem in this paper we address the anti windup design problem for linear control systems with strictly proper controllers in the presence of input magnitude and rate saturation using generalized sector condition we provide an lmi based procedure for the construction of a linear anti windup gain acting on the controller state equation such that regional closed loop stability is guaranteed and suitable performance measures are optimized the approach is successfully illustrated on a simulation example

trichotomy for finitely generated monomial algebras abstract it is proved that a finitely generated prime monomial algebra either is primitive or it satisfies a polynomial identity or it is not semiprimitive this confirms a conjecture of bell and smoktunowicz

noncommutative graded domains with quadratic growth letk be an algebraically closed field and letr be a finitely generated connected gradedk algebra which is a domain of gelfand kirillov dimension two write the graded quotient ringq r ofr asd z z 1 for some automorphism of the division ringd we prove thatd is a finitely generated field extension ofk of transcendence degree one moreover we describer in terms of geometric data ifr is generated in degree one then up to a finite dimensional vector space r is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curvey this implies in particular thatr is noetherian thatr is primitive when and thatr is a finite module over its centre when ifr is not generated in degree one thenr will still be noetherian and primitive if has infinite order butr need not be noetherian when has finite order

finite time h control for a class of discrete time switched singular time delay systems subject to actuator saturation this paper concerns with the finite time h control problem for a class of discrete time switched singular time delay with actuator saturation not only linear matrix inequality conditions for the systems but also average dwell time of switching signal is given to guarantee regular causal and finite time boundedness for the discrete time switched singular time delay system moreover sufficient conditions are presented to ensure the h disturbance attenuation level and the design method of h controller is developed by solving linear matrix inequalities lmi optimization problem without any decompositions of system matrices and equivalent transformation furthermore the function in the proof procedure belongs to multiple lyapunov like functions whose advantage lies in their flexibility finally numerical examples are employed to verify the effectiveness of the proposed methods and to illustrate the significant improvement on the conservativeness of some reported results in the literature

robust finite time h control for discrete time singular markovian jump systems with time varying delay and actuator saturation in this paper the problem of robust finite time h control for discrete time singular markovian jump systems with time varying delay and actuator saturation is studied first sufficient conditions that guarantee the systems singular stochastic finite time boundedness are derived via constructing a delay dependent lyapunov krasonskii functional and linear matrix inequalities lmi approaches then the results are extended to singular stochastic finite time h boundedness furthermore with these conditions and convex optimization arithmetic the design method of h controller and conditions that ensure the h disturbance attenuation level are obtained finally some numerical examples are presented to illustrate the validity and efficiency of the proposed methods

improvements to the linear differential inclusion approach to stability analysis of linear systems with saturated linear feedback ellipsoids as level sets of quadratic lyapunov functions and the convex hull of ellipsoids as a level set of a certain composite quadratic lyapunov function have both been extensively used as estimates of the domain of attraction of a linear system under saturated linear feedback by expressing the saturated linear feedback law on the convex hull of a group of linear feedback laws which in turn expresses the linear system under this saturated linear feedback in a linear differential inclusion conditions have been established under which an ellipsoid or the convex hull of a group of ellipsoids are contractively invariant sets and are thus estimates of the domain of attraction these conditions are usually less conservative for single input systems than for multiple input systems in this paper we consider multiple input systems and establish conditions for contractive invariance of the convex hull of ellipsoids that are less conservative than the existing conditions

absolute stability analysis of discrete time systems with composite quadratic lyapunov functions a generalized sector bounded by piecewise linear functions was introduced in a previous paper for the purpose of reducing conservatism in absolute stability analysis of systems with nonlinearity and or uncertainty this paper will further enhance absolute stability analysis by using the composite quadratic lyapunov function whose level set is the convex hull of a family of ellipsoids the absolute stability analysis will be approached by characterizing absolutely contractively invariant aci level sets of the composite quadratic lyapunov functions this objective will be achieved through three steps the first step transforms the problem of absolute stability analysis into one of stability analysis for an array of saturated linear systems the second step establishes stability conditions for linear difference inclusions and then for saturated linear systems the third step assembles all the conditions of stability for an array of saturated linear systems into a condition of absolute stability based on the conditions for absolute stability optimization problems are formulated for the estimation of the stability region numerical examples demonstrate that stability analysis results based on composite quadratic lyapunov functions improve significantly on what can be achieved with quadratic lyapunov functions

the maximal contractively invariant ellipsoids for discrete time linear systems under saturated linear feedback abstract in this paper we consider the problem of determining the maximal contractively invariant ellipsoids for discrete time linear systems with multiple inputs under saturated linear feedback we propose an algebraic computational approach to determining such maximal contractively invariant ellipsoids we divide the state space into several regions according to the saturation status of each input and compute the possible maximal contractively invariant ellipsoids on each region except the region where none of inputs saturate and on their intersections the minimal one among these possible maximal contractively invariant ellipsoids is the maximal contractively invariant ellipsoids of the system simulation results demonstrate the effectiveness of our methods

zariski cancellation problem for non domain noncommutative algebras we study zariski cancellation problem for noncommutative algebras that are not necessarily domains

star shaped control vector sets of second order systems with pwm type input this paper deals with second order continuous time linear time invariant systems by means of a pulse width modulation pwm type input in general it is difficult to analyze and design a pwm system because of its nonlinearity in terms of the duty ratio in this paper it is shown that the nonlinearity appearing in the discrete time model at the sampling instants can be eliminated by varying the location of the pulse as well as the duty ratio

design of linear systems with saturating linear control and bounded states a theorem is developed based on the circle criterion which gives sufficient conditions for the stability of a saturating linear control for a linear system with bounded states this theorem can be applied to determine a range of gains for the linear control which ensures stability it is shown by example that this approach results in an increased speed of response over that obtained by previous methods which do not account for state bounds

dynamic output feedback for discrete time systems under amplitude and rate actuator constraints this work proposes a technique for the design of stabilizing dynamic output feedback controllers for discrete time linear systems with rate and amplitude saturating actuators the nonlinear effects introduced by the saturations in the closed loop system are taken into account by using a generalized sector condition which leads to theoretical conditions for solving the problem directly in the form of linear matrix inequalities

robust controller design of uncertain discrete time delay systems with input saturation and disturbances this technical note deals with the problem of robust controller design of uncertain discrete time delay systems subject to control input saturation and bounded external disturbances a sufficient condition is obtained which guarantees the existence of state feedback controllers and an admissible initial condition domain such that the resulting closed loop system is uniformly exponentially convergent to a ball with certain exponential decay rate for every initial condition emanating from the admissible domain an iterative linear matrix inequality relaxation scheme can be used to construct desired state feedback controllers a simulation example is provided to demonstrate the effectiveness of the proposed method

representations of solvable lie algebras v on the gelfand kirillov dimension of simple modules abstract examples are given of simple noetherian integral domains which have simple modules of widely differing gk dimensions there is also an example of a simple integral domain whose gk dimension is one

classification of affine prime regular hopf algebras of gk dimension one abstract the classification of affine prime regular hopf algebras of gk dimension one is completed as consequences 1 we give a negative answer to question 7 1 posed in 5 and 2 we show that there do exist prime regular hopf algebras of gk dimension one which are not pointed

invariant sets computation for convex difference inclusions systems in this paper we introduce the convex difference inclusion cdi systems as a modeling framework useful to analyze set theory and invariance related issues for nonlinear and uncertain systems the dynamics of a cdi system is given by a set valued map whose values are convex compact subsets of the space and are determined by convex bounding functions necessary and sufficient boundary type conditions for invariance and contractiveness characteristic of the linear systems are given for the cdi systems lyapunov functions are proved to be induced by contractive sets for cdi systems as in the linear context a computational procedure for obtaining polytopic invariant and contractive sets for nonlinear systems based on the properties of the cdi systems is presented

linear algebra approach to neural associative memories and noise performance of neural classifiers the authors present an analytic evaluation of saturation and noise performance for a large class of associative memories based on matrix operations the importance of using standard linear algebra techniques for evaluating noise performance of associative memories is emphasized the authors present a detailed comparative analysis of the correlation matrix memory and the generalized inverse memory construction rules for auto associative memory and neural classifiers analytic results for the noise performance of neural classifiers that can store several prototypes in one class are presented the analysis indicates that for neural classifiers the simple correlation matrix memory provides better noise performance than the more complex generalized inverse memory

discrete time l and l2 norm vanishment and low gain feedback with their applications in constrained control the existing low gain feedback which is a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero has been designed in very specific ways in this paper by recognizing the l and l 2 slow peaking phenomenon that exists in discrete time systems under low gain feedback more general notions of l and l 2 norm vanishment are considered so as to provide a full characterization of the nonexistence of slow peaking phenomenon in some measured signals low gain feedback that does not lead to l and l 2 slow peaking in the control input are respectively referred to as l and l 2 low gain feedback based on the notions of l and l 2 vanishment not only can the existing low gain feedback be recognized as an l low gain feedback but also a new design approach referred to as the l 2 low gain feedback approach is developed for discrete time linear systems parallel to the effectiveness of l low gain feedback in magnitude constrained control the l 2 low gain feedback is instrumental in the control of discrete time systems with control energy constraints the notions of l and l 2 vanishment also result in a systematic approach to the design of l and l 2 low gain feedback by providing a family of solutions including those resulting from the existing design methods

robust control of uncertain discrete time markovian jump systems with actuator saturation in this paper the robust stochastic stabilization problem for the class of discrete time uncertain markovian jump linear systems mjls with actuator saturation is considered the definition of domain of attraction in mean square sense doa mss is introduced to analyze the stochastic stability of the closed loop system by using a class of stochastic lyapunov function which is dependent on the jump mode and saturation function design procedures for both the mode dependent and mode independent state feedback controllers are developed based on the linear matrix inequality lmi approach finally a numerical example is provided to show the usefulness of the proposed techniques

on structure of domains with quadratic growth abstract let f be a finite field and let r be an affine f algebra which is a domain of gelfand kirillov dimension smaller than 3 let m n be natural numbers assume that x r is transcendental over f and y 1 y n r are such that i j m i j x i y k x j 0 for some i j f not all equal to 0 and each k n it is shown that either r satisfies a polynomial identity or else the subalgebra of r generated by y 1 y 2 y n and x has gelfand kirillov dimension 1 from this we deduce that a finitely generated domain over f with quadratic growth and with an infinite centre satisfies a polynomial identity is a pi domain moreover the centralizer of a non algebraic element in a finitely generated domain with quadratic growth over finite field is a pi domain

l 2 gain analysis and control synthesis of uncertain discrete time switched linear systems with time delay and actuator saturation the problem of l 2 gain analysis and control synthesis is studied for a class of uncertain discrete time switched linear systems with time delay and saturating actuators by using the multiple lyapunov functions method with the state feedback controllers adopted beforehand an analysis condition on disturbance tolerance is derived under which the operating state trajectory with zero initial condition will remain inside a bounded set in its proximity then with this condition at hand the largest disturbance tolerance level is determined via the solution of a constrained optimisation problem then a sufficient condition for the restricted l 2 gain over the set of tolerable disturbances is derived the smallest upper bound on the restricted l 2 gain is also obtained by solving a constrained optimisation problem furthermore when controller gain matrices are design variables these optimisation problems are adjusted for solving the control design task an illustrative numerical example is given to demonstr

structure of prime finitely presented monomial algebras abstract the structure of a finitely presented monomial algebra k x k i over a field k is described here x is a finitely generated free monoid and i is a prime ideal of x that is finitely generated as an application a new structural proof of the recent result of bell and pekcagliyan j bell p pekcagliyan primitivity of finitely presented monomial algebras preprint arxiv 0712 0815v1 on the primitivity of such algebras is presented which yields a positive solution to the trichotomy problem raised by bell and smoktunowicz j bell a smoktunowicz the prime spectrum of algebras of quadratic growth j algebra 319 2008 414 431 in the finitely presented case our approach is based on a new result on the form of prime rees factors of semigroups satisfying the ascending chain condition on one sided annihilators and on its refinement in the case of finitely presented factors of the form x i

approximation and monotonicity of the maximal invariant ellipsoid for discrete time systems by bounded controls an analytic approximation of the maximal invariant ellipsoid for a discrete time linear system with bounded controls is derived the approximation is expressed explicitly in terms of the coefficient matrices of the system and the positive definite matrix that represents the shape of the invariant ellipsoid it is shown that this approximation is very close to the exact maximal invariant ellipsoid obtained by solving either an lmi based optimization problem or a nonlinear algebraic equation furthermore the necessary and sufficient condition for such an approximation to be equal to the exact maximal invariant ellipsoid is established on the other hand the monotonicity of the maximal invariant ellipsoid resulting from the minimal energy control with guaranteed convergence rate problem is established that shows a trade off between increasing the size of the invariant ellipsoid and increasing the convergence rate of the closed loop system under a bounded control two illustrative examples demonstrate of the effectiveness of the results