linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

optimal interpolation for linear stochastic systems let y satisfying dy cxdt ddw y 0 0 represent the noisy observation of the gauss markov process x generated by dx axdt bdw x 0 x 0 on the finite interval 0 t in this paper we solve the following interpolation problem determine the best least squares estimate of x t t in 0 t given the increments of y on the intervals 0 t 1 and t 2 t where 0 t 1 t 2 t and the statistics of x 0 several alternative expressions for the optimal interpolator are derived some of which are phrased in terms of two kalman bucy estimates as a by product we also solve the interpolation problem for the missing increments of y

asymmetricalgebraicriccatiequation ahomeomorphicparametrizationofthesetof solutions abstract in this paper asymmetric algebraic riccati equations are analyzed in particular we derive a new parametrization of the set of solutions generalizing on the symmetric case the proposed parametrization is obtained in terms of pairs of invariant subspaces of two related feedback matrices moreover the connection is clarified between the new parametrization and the classical homeomorphic one based on graph invariant subspaces of the pseudo hamiltonian matrix associated with the equation we finally show that also the newly introduced parametrization is given by a homeomorphic map

estimating the plant state from the compensator state for the system dot w sw where w x over z this note states and proves necessary and sufficient conditions under which a z is an estimate of a linear function of x i e z estimates tx or b a linear function of z is an estimate of x i e vz estimates x the mapping t has been used previously by the authors to compute the estimator controller form of a compensator whose state vector z had the same dimension as the plant state vector x

on the relation between additive and multiplicative decompositions of rational matrix functions in this paper a general frame to study relations between additive and multiplicative representations of rational matrix functions is presented various extensions of the positive real lemma as well as of other classical factorization results both in the continuous and discrete time cases are established in the case of square factorizations the map between solutions to an asymmetric algebraic riccati equation and pairs of factors is shown to be a homeomorphism in this framework we also derive a geometric characterization of non square factorizations

generation of finite classical groups by pairs of elements with large fixed point spaces abstract we study good elements in finite 2 n dimensional classical groups g namely t is a good element if o t is divisible by a primitive prime divisor of q n 1 for the relevant field order q and t fixes pointwise an n space the group sl 2 n q contains such elements and they are present in su 2 n q sp 2 n q so 2 n e q only if n is odd even even respectively we prove that there is an absolute positive constant c such that two random conjugates of t generate g with probability at least c if g so 2 n e 2 and g sp 2 n q with q even in the exceptional case g sp 2 n q with q even two conjugates of t never generate g in this case we prove that two random conjugates of t generate a subgroup so 2 n e q with probability at least c the results underpin analysis of new constructive recognition algorithms for classical groups in even characteristic which succeed where methods utilising involution centralisers are not available

bounds for the minimum numbers of generators of generalized cohen macaulay ideals throughout this paper r denotes a local ring with maxima1 idea1 m z an ideal of r and s the factor ring r z we call z a generalized cohen macaulay idea1 if s is a generalized cohen macaulay ring by definition 4 3 1 a local ring a with maxima1 ideal is a generalized cohen macaulay ring if the ith local cohomology module h a of a with support b is of finite length for i o dim a 1 a is a cohen macaulay ring iff h a 0 for i o dim a 1 the class of generalized cohen macaulay local rings is rather large for instance if a is a factor ring of a cohen macaulay ring a being a generalized cohen macaulay ring means that a is equidimensional and a is a cohen macaulay ring for all prime ideals y of a 4 2 5 and 381 the aim of this paper is to give bounds for the minimum number u z of generators of a generalized cohen macaulay idea1 z in terms of the following invariants of r and s m u m n v ms r dim r s dim s the multiplicities e r e s and z zyzl s i o s 1 where 1 denotes the length function our interest in such bounds for u z arose from the article 7 of sally there she proved that if r and s are cohen macaulay rings then v z e r e s r s l r s 1 first sally s result may be generalized as

convergence of the time invariant riccati differential equation and lq problem mechanisms of attraction the nature of the attraction of the solution of the time invariant matrix riccati differential equation towards the stabilizing solution of the algebraic riccati equation is studied this is done on an explicit formula for the solution when the system is stabilizable and the hamiltonian matrix has no eigenvalues on the imaginary axis various aspects of this convergence are analysed by displaying explicit mechanisms of attraction and connections are made with the literature the analysis ultimately shows the exponential nature of the convergence of the solution of the riccati differential equation and of the related finite horizon lq optimal state and control trajectories as the horizon recedes computable characteristics are given which can be used to estimate the quality of approximating the solution of a large finite horizon lq problem by the solution of an infinite horizon lq problem

computing the estimator controller form of a compensator a technique is presented for finding the estimator and controller gains to implement a desired compensator in estimator controller form the technique is derived by mapping the states of the compensator to those of the estimator controller via a linear transformation using the technique involves finding the real invertible solutions of a non symmetric steady state matrix riccati equation the ricatti equation may have zero two or more real invertible solutions from each solution a set of estimator and controller gains can be computed multiple solutions correspond to the freedom to choose which closed loop system poles are associated with the controller and which are associated with the estimator this technique can be applied to multi input multi output systems in an example the technique is applied to the despin control system of a dual spin satellite

singular perturbation modelling of large scale systems with multi time scale property the problem is investigated of singular perturbation modelling which plays an important role in cases of the application of the singular perturbation method to the design and analysis of large scale systems in both continuous and discrete time versions modelling concepts based on structural properties are studied and computer oriented algorithms are developed the results of two time scale systems are extended to multi time scale cases furthermore in order to verify the validity of these studies some numerical examples are given

on liaison arithmetical buchsbaum curves and monomial curves in p3 in the context of liaison by 181 and 191 for curves in ip3 the property of being the zero scheme of a section of a rank two bundle on ip3 is connected to that of being ideally the intersection of three surfaces an immediate consequence of this property is that the homogeneous ideal i c of a curve c c p3 is generated by precisely three elements if and only if c is ideally the intersection of three suraces and c is arithmetically cohen macaulay non complete intersection furthermore the characterization of the equivalence classes defined by liaison among curves in p3 yields the well known statement that a curve c is in the liaison class of a complete intersection if and only if c is arithmetically cohen macaulay see e g 2 10 13 or 141 it seems that a cayley see 7 p 1521 in 1847 was the first who posed the problem to describe this liaison class of a complete intersection now in this paper we will begin to investigate the next simple case that is the liaison classes which are characterized by a finite dimensional vector space of dimension l using the theory of buchsbaum rings see section 2 this means we will study liaison among arithmetical buchsbaum curves in p3 first from the point of view of local algebra we get the following more general result

on the phase portrait of the matrix riccati equation arising from the periodic control problem a comprehensive description is provided of the properties of the matrix riccati differential equation in which the coefficient matrices are t periodic for some t 0 the main results include 1 a classification of all the periodic equilibria t periodic solutions 2 necessary and sufficient conditions for convergence to the uniformly asymptotically stable completely unstable periodic equilibrium as t to infty t to infty 3 necessary and sufficient conditions for a solution to escape in finite forward or backward time 4 a description of every almost periodic solution 5 a proof that every solution which does not escape in finite time is asymptotically almost periodic and an explicit formula for the limiting almost periodic solution

necessary conditions for constant solutions of coupled riccati equations in nash games abstract the asymptotic behaviour of solutions as t for coupled matrix riccati equations occurring in open loop linear quadratic nash games is studied in this paper a general formula representing all possible solutions is given necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained two examples illustrate these results

constructive recognition of classical groups in even characteristic abstract let g x gl d f be a classical group in its natural representation defined over a finite field f of even characteristic we present las vegas algorithms to construct standard generators for g which permit us to write an element of g as a straight line program in x and to construct an involution as a straight line program in x if f 4 then the algorithms run in time polynomial in the size of the input subject to the existence of a discrete logarithm oracle for f

new method for optimal control and filtering of weakly coupled linear discrete stochastic systems abstract the algebraic regulator and filter riccati equations of weakly coupled discrete time stochastic linear control systems are completely and exactly decomposed into reduced order continuous time algebraic riccati equations corresponding to the subsystems that is the exact solution of the global discrete algebraic riccati equation is found in terms of the reduced order subsystem nonsymmetric continuous time algebraic riccati equations in addition the optimal global kaiman filter is decomposed into local optimal filters both driven by the system measurements and the system optimal control inputs as a result the optimal linear quadratic gaussian control problem for weakly coupled linear discrete systems takes decomposition and parallelism between subsystem filters and controllers

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

geometric aspects of the riccati difference equation in the nonsymmetric case abstract this paper provides a collection of geometric results concerning the nonsymmetric riccati difference equation it gives representation formulas which express all the solutions in terms of a suitable number of known ones further results include geometric properties of the difference of two solutions and the generalization of some of the author s earlier results to the nonsymmetric case

pole placement by performance criterion modification a procedure is presented that modifies the state weighting matrix q and introduces a degree of relative stability into the original performance criterion to shape the resulting optimal dynamics by positioning the closed loop eigenvalues along the real axis of the optimal system in the linear quadratic regulator problem it is based on the results of the algebraic riccati equation are which establish the invariance of certain eigenspaces of the associated hamiltonian matrix with respect to certain perturbations of q and the degree of relative stability

invariant manifolds and projective combinations of solutions of the riccati differential equation abstract in this paper we show how families of solutions of the general riccati differential equation rde can be generated via projective combinations of a given number of reference solutions our approach is based upon the extension of the domain of the equation to the grassmannian manifold and the application of the radon lemma in this context we briefly discuss the relevance of our results to the study of the invariant manifolds of the equation and compare them to existing results concerning representation formulas for solutions of rde the results of the paper have been motivated by the recent characterization of solutions of the rde given in m pavon d d alessandro families of solution of matrix riccati differential equations siam j control optim 35 1 1997 194 204 which extends the classical results on the algebraic riccati equation due to willems coppel and shayman

robust control with pole shifting via performance index modification robust controllers that ensure closed loop eigenvalue placement are designed for linear systems with parametric uncertainties by using an appropriate state weighting matrix choice and a relative stability degree an lmi based algorithm is proposed to provide guaranteed cost controllers that ensure i the closed loop nominal eigenvalues shifting by a pre specified amount ii a prescribed degree of stability for all system uncertainties of a given class a certain level of closed loop performance is also ensured by minimizing the associated guaranteed cost

on equations defining arithmetically cohen macaulay schemes i in this paper we continue the study of the simplest arithmetically cohen macauly cm for short schemes cf 111 and 12 i we describe the hilbert function of an arbitrary cm scheme in p of given degree e lying on a hypersurface of given degree k and such that its general curvilinear section has minimal genus proposition 1 3 we present a geometric construction of all reduced irreducible codimension two cm varieties whose general curvilinear sections have minimal genus with respect to degree theorem 1 7 we also prove the following assertion essentially due to severi there exist smooth nonspecial cm curves of degree n and genus g in p iff r y1 g r d t y 1 2 proposition 1 6 any such curve has minimal genus with respect to degree in the last section we consider d dimensional cm schemes of degree e mvd j jed in p whose general curvilinear sections have minimal genus these are the most accessible cm schemes cf 12 i we prove that the corresponding part of the hilbert scheme is a connected smooth variety naturally birationally equivalent to nd symemvd p provided v d 2 theorem 2 6 this was known for m v d 2 3 1 cf 9 our proof relies on an explicit description of the infinitesimal deformations of a thick subspace of arbitrary codimension see 2 4 we work over an algebraically closed field of characteristic 0 by the genus of a cm curve we mean its arithmetic genus

geometric properties and invariant manifolds of the riccati equation this short paper considers the general riccati matrix differential equation it reviews and extends results on the characterization and existence of equilibrium solutions establishes that the riccati equation has at most one stable equilibrium solution as t rightarrow infty or t rightarrow infty and confines the region of attraction to this unique stable equilibrium solution by identifying and utilizing linear and nonlinear invariant manifolds of the riccati equation

periodic riccati difference equations approaching equilibria by implicit systems a parameterization of solutions to the periodic riccati difference equation rde is given by using behavioral tools the moving equilibria of the periodic rde are encompassed among the trajectories of the implicit ar model associated to the symplectic pencil of the corresponding hamiltonian equations this allows a parameterization of solutions of periodic rdes even in the case the dynamic matrix is singular

the exact slow fast decomposition of the algebraic ricatti equation of singularly perturbed systems the algebraic riccati equation for singularly perturbed control systems is completely and exactly decomposed into two reduced order algebraic riccati equations corresponding to the slow and fast time scales the pure slow and pure fast algebraic riccati equations are asymmetric ones but their o epsilon perturbations are symmetric it is shown that the newton method is very efficient for solving the obtained asymmetric algebraic riccati equations the method presented is very suitable for parallel computations due to the complete and exact decomposition of the riccati equation this procedure might produce new insight into the two time scale optimal filtering and control problems

families of solutions of matrix riccati differential equations the j c willems coppel shayman geometric characterization of solutions of the algebraic riccati equation are is extended to asymmetric riccati differential equations with time varying coefficients the coefficients do not need to satisfy any definiteness periodicity or system theoretic condition more precisely given any two solutions x 1 t and x 2 t of such equation on a given interval t 0 t 1 we show how to construct a family of solutions of the same equation of the form x t i pi t x 1 t pi t x 2 t where pi is a suitable matrix valued function even when specialized to the case of x 1 and x 2 equilibrium solutions of a symmetric equation with constant coefficients our results considerably extend the classical ones as no further assumption is made on the pair x 1 x 2 and on the coefficient matrices

upper and lower bounds for the solution of the general matrix riccati differential equation on a time scale we obtain upper and lower bounds for the solution of the general matrix riccati differential equation on a time scale t r t a t b t r t r t b t r t c t r t where a t and c t are symmetric n n matrices while b t v t t t and r t are n n matrices and denotes the transpose of the matrix we use the quasilinearization technique to obtain these bounds we also study the monotonicity of the successive approximations

a survey of nonsymmetric riccati equations abstract we survey recent and also older results on nonsymmetric matrix riccati differential equations and in the time invariant case on the corresponding algebraic riccati equations in particular we cite various applications connected with matrix riccati equations

brief paper eigenvector approach for order reduction of singularly perturbed linear quadratic optimal control problems in this paper we show how to decompose the singularly perturbed algebraic riccati equation and the corresponding linear quadratic optimal control problem at steady state in terms of reduced order pure slow and pure fast problems by using the eigenvector approach the eigenvector approach should be used for decomposition of singularly perturbed control systems in the cases when the singular perturbation parameter is not very small in such cases the decomposition methods based on series expansions fixed point iterations subspace iterations and newton iterations either fail to produce solutions of the corresponding algebraic equations or display very slow convergence

growth of cross characteristic representations of finite quasisimple groups of lie type abstract in this paper we give a bound to the number of conjugacy classes of maximal subgroups of any almost simple group whose socle is a classical group of lie type the bound is 2 n 5 2 n log 2 log 2 q where n is the dimension of the classical socle and q is the size of the defining field to obtain the bound we first bound the number of projective cross characteristic representations of simple groups of lie type as a function of the representation degree these bounds are computed for different families of groups separately in the computation we use information on conjugacy class numbers minimal character degrees and gaps between character degrees

effective black box constructive recognition of classical groups abstract we describe a black box las vegas algorithm to construct standard generators for classical groups defined over finite fields we assume that the field has size at least 4 and that oracles to solve certain problems are available subject to these assumptions the algorithm runs in polynomial time a practical implementation of our algorithm is distributed with the computer algebra system magma