exact pole assignment using direct or dynamic output feedback this note addresses the problem of the assignability of the eigenvalues of the matrix a bpc by choice of the matrix p this mathematical problem corresponds to pole assignment in the direct output feedback control problem and by proper changes of variables it also represents the pole assignment problem with dynamic feedback controllers the key to our solution is the introduction of the new concept of local complete assignability which in loose terms is the arbitrary perturbability of the eigenvalues of a bpc by perturbations of p if n x is the order of the system we show that if a bp 0 c has distinct eigenvalues a necessary and sufficient condition for local complete assignability at p 0 is that the matrices c a bp 0 c i 1 b be linearly independent for 1 leq i leq n x in special cases this condition reduces to known criteria for controllability and observability although these latter properties are necessary conditions for assignability we also address the question of the assignability of uncontrollable or unobservable systems both by direct output feedback and dynamic compensation the main result of this note yields an algorithm that assigns the closed loop poles to arbitrarily chosen values in the direct and in the dynamic output feedback control problems

gain optimization under control structure and stability region constraints in this paper we consider the problem of optimizing the frobenious norm of linear static feedback gains of a system with control structure and stability region constraints we consider four important control structure constraints these are i static output feedback ii constant gain feedback for multiple operating conditions iii two controller feedback and iv decentralized feedback we obtain a nonlinear programming formulation for the optimization problem and present an effective numerical approach for solving it we include four examples

when is a regular local ring a locality let a be a regular local ring and let r be a polynomial extension of a in a finite number of variables several attempts have been made to study the structure of projective modules over r and to find the minimal numbers of generators of ideals of r to wit z v 4 3 61 the methods employed in these references depend very much on the structure of a the cases dealt with are formal power series rings over fields and local rings at smooth points of algebraic varieties cohen s structure theorem tells us precisely what regular local rings are formal power series rings we present here the following characterization for a regular local ring to be a locality over a field

a computational algorithm for pole assignment of linear single input systems an efficient computational algorithm for pole assignment of linear multiinput systems is proposed a preliminary stage of the algorithm is a reduction of the system matrices into orthogonal canonical form the gain matrix elements are then found by orthogonal transformation of the closed loop system matrix into upper quasi triangular form whose diagonal blocks correspond to the desired poles the algorithm is numerically stable the computed gain matrix being exact for a system with slightly perturbed matrices it works equally well with real and complex distinct and multiple poles and is applicable to ill conditioned and high order problems

use of the design freedom of time optimal control abstract a method for synthesizing a state feedback time optimal controller for linear n state r input discrete time systems is developed the nonuniqueness of the solution is organized in such a way as to select the controller with the minimum value of the frobenius norm of the feedback gain matrix k it is shown that among the rn design parameters in the parametric form of k only r n rp r need to be considered in the search technique where p r is the order of the smallest jordan block in the jordan form of the closed loop system

generalized fitting modules and rings abstract an r module m is called strongly hopfian respectively strongly co hopfian if for every endomorphism f of m the chain ker f ker f 2 respectively im f im f 2 stabilizes the class of strongly hopfian respectively co hopfian modules lies properly between the class of noetherian respectively artinian and the class of hopfian respectively co hopfian modules for a quasi projective respectively quasi injective module m if m is strongly co hopfian respectively strongly hopfian then m is strongly hopfian respectively strongly co hopfian as a consequence we obtain a version of hopkins levitzki theorem for strongly co hopfian rings namely a strongly co hopfian ring is strongly hopfian also we prove that for a commutative ring a the polynomial ring a x is strongly hopfian if and only if a is strongly hopfian

on the relationship between controllability indexes eigenvector assignment and deadbeat control the subspaces to which closed loop generalized eigenvectors are restricted are described in terms of the controllability indexes of the pair a b and the polynomials of minimal degree that span ker a lambda i b this characterization of the eigenspaces is then used to calculate the deadbeat controller of minimum frobenius norm

some isolation results for f harmonic maps on weighted riemannian manifolds with boundary abstract in this paper we study f harmonic respectively p f harmonic maps on a compact weighted riemannian manifold of nonempty boundary and of positive bakry emery ricci curvature we first establish a bochner reilly formula for such maps and deduce therefrom some immediate isolation results in addition using a weighted sobolev inequality obtained in 12 we prove that under some energy level depending on the bakry emery curvature of the initial manifold and on the sectional curvature of the final one the only f harmonic respectively p f harmonic maps are the constant ones a gap property of the energy density of f harmonic maps from a weighted riemannian manifold to a unit sphere is also obtained

homogeneous einstein metrics and simplicial complexes on compact homogeneous spaces we investigate the hilbert action restricted to the space of homogeneous metrics of volume one based on a detailed understanding of the high energy levels of this action we assign to a compact homogeneous space a simplicial complex whose non contractibility is a sufficient condition for the existence of homogeneous einstein metrics

minimum norm output feedback design under specified eigenvalue areas abstract starting from a new parametric expression for the state feedback controller of a linear time invariant system a procedure is derived which reduces the design of output feedback controllers to an unconstrained optimization problem where all free parameters both the closed loop eigenvalues and the invariant parameter vectors are used for its numerical solution by a gradient based search technique thus minimal norm output feedback controllers can be designed that place the closed loop eigenvalues within specified regions of the eigenvalue plane

fine structure of the space of spherical minimal immersions the space of congruence classes of full spherical minimal immersions f sm sn of a given source dimension m and algebraic degree p is a compact convex body mp in a representation space fp of the special orthogonal group so m 1 in ann of math 93 1971 43 62 docarmo and wallach gave a lower bound for fp and conjectured that the estimate was sharp toth resolved this exact dimension conjecture positively so that all irreducible components of fp became known the purpose of the present paper is to characterize each irreducible component v of fp in terms of the spherical minimal immersions represented by the slice v n mp using this geometric insight the recent examples of deturck and ziller are located within mp

universal constraints on the range of eigenmaps and spherical minimal immersions the purpose of this paper is to give lower estimates on the range dimension of spherical minimal immersions in various settings the estimates are obtained by showing that infinitesimal isometric deformations with respect to a compact lie group acting transitively on the domain of spherical minimal immersions give rise to a contraction on the moduli space of the immersions and a suitable power of the contraction brings all boundary points into the interior of the moduli space 1 preliminaries and statement of results an isometric immersion f s sv m 2 of the euclidean mr sphere sk of constant curvature k into the unit sphere sv of a euclidean vector space v is said to be a spherical minimal immersion if f is minimal by a result of takahashi 11 f exists if k m ap for some p 1 where ap p p m1 is the p th eigenvalue of the spherical laplacian as7m on sm s and in this case each component o f q e v of f is a spherical harmonic of order p on sm an eigenfunction of asm with eigenvalue ap we scale the metric on the domain to curvature one and call f sm s sv a spherical minimal immersion of algebraic degree p because of the scaling f is a homothetic minimal immersion with homothety ap m 1 f x x f y ap m xi y where x y are vector fields on sm let hp denote the euclidean vector space of spherical harmonics of order p on sm endowed with the l2 scalar product suitably scaled the universal example of a spherical minimal immersion is the standard minimal immersion fm p sm shp whose components relative to an orthonormal basis in hp are orthonormal for p 2 fm 2 sm s 2 is the classical veronese map unless relevant we suppress m from the notation and write the standard minimal immersion as fp sm sep a fundamental problem raised by m docarmo and n wallach 3 is to find lower bounds for the range dimension of spherical minimal immersions the main result of this paper theorem 4 solves this problem by giving lower bounds in terms of the dimension of the domain the degree and differential geometric properties of the immersions such as symmetries and higher order isotropy the only previously known general lower bound was given by j d moore 10 who proved that for a spherical minimal immersion f sm sv of degree 2 we have dim v 2m docarmo and wallach conjectured 3 that the lower bound received by the editors april 20 1997 1991 mathematics subject classification primary 53c42 1999 american mathematical society

a convergence theorem for riemannian submanifolds in this paper we study the convergence of riemannian submanifolds in particular we prove that any sequence of closed submanifolds with bounded normal curvature and volume in a closed riemannian manifold subconverge to a closed submanifold in the c1 a topology we also obtain some applications to irreducible homogeneous manifolds and pseudo holomorphic curves in symplectic manifolds

on a question of quillen let r be a regular local ring and f a regular parameter of r quillen asked whether every projective r f module is free we settle this question when r is a regular local ring of an affine algebra over a field k further if r has infinite residue field we show that projective modules over laurent polynomial extensions of r f are also free

paper transfer matrix approach to the triangular block decoupling problem the triangular block decoupling problem for linear multivariable systems is studied via the transfer matrix approach this approach clearly separates the problem of admissible control laws from the one of desired decoupled system specifications necessary and sufficient triangular decoupling conditions are given for various control laws these conditions are expressed in a very simple way in terms of linear dependance among the transfer matrix rows it turns out that when the problem is solvable this can be done by static state feedback on a minimal realization of the system furthermore it is shown that whenever triangular block decoupling is possible it is also attainable with stability

brief paper a continuation approach to eigenvalue assignment this paper describes a continuation approach to eigenvalue assignment the method is a homotopy technique which embeds the control problem into a parameterized family of control problems this parameterization describes a continuous deformation of a system with the desired spectrum into the original system based on this deformation a differential equation is constructed whose solution trajectory has an end point which is a constant output feedback matrix assigning the desired spectrum to the original system the derivation of this differential equation and conditions which guarantee the existence of a solution are given also two examples of its numerical implementation are detailed

minimal isometric immersions of inhomogeneous spherical space forms into spheres a necessary condition for existence although much is known about minimal isometric immersions into spheres of homogeneous spherical space forms there are no results in the literature about such immersions in the dominant case of inhomogeneous space forms for a large class of these we give a necessary condition for the existence of such an immersion of a given degree this condition depends only upon the degree and the fundamental group of the space form and is given in terms of an explicitly computable function evaluating this function shows that neither l 5 2 nor l 8 3 admit a minimal isometric immersion into any sphere if the degree of the immersion is less than 28 or less than 20 respectively

on triviality of the euler class group of a deleted neighbourhood of a smooth local scheme let k be an infinite perfect field and let x specr where r is the localization of a k algebra b of finite type with respect to a regular prime ideal p specb letd be a divisor onx with a decompositiond i 1di into its irreducible components di such that each di is nonsingular and they intersect transversally consider y x d a deleted neighbourhood of the closed point x0 of x let e be a vector bundle on y in this set up e is trivial in the following cases i dim x 3 gabber ga1 ii t 1 bhatwadekar rao br iii ranke min dimr 2 t rao ra1 the above exposition is borrowed from a paper of nisnevich ni which contains some beautiful applications of these purity theorems in the same set up now let y y be a closed point and let my be the corresponding maximal ideal of oy y it is natural to ask whether my is a complete intersection observing that the conormal bundle my my is trivial we consider a more general situation let i be the defining ideal of a closed subscheme z of y such that the conormal bundle i i2 defined on z is trivial is i a complete intersection the questions we are addressing in this paper are a little stronger in nature can a given basis of i i2 be lifted to a minimal set of generators of i we remark that with the hypothesis on x each component di is defined by a single equation fi which vanishes at the closed point x0 of x let m be the maximal ideal at x0 the condition of transversality of intersection of di s is equivalent to saying that the images of f1 ft in the a m vector space m m2 are linearly independent keeping this discussion in mind we now give an excerpt of our main results in commutative algebraic terms see 4 2 4 3 4 4

feedback controls for pole subset assignment a pole assignment problem for multi input linear systems is discussed in conventional pole assignment problems an input is concerned with all poles simultaneously or it assigns only a specified subset of poles here we propose a design method which makes each input assign an arbitrary subset of poles thus when we replace a pole by using this method it is enough to reconstruct only the corresponding input

infinitesimal rotations of isometric minimal immersions between spheres a fundamental problem posed by docarmo and wallach is to give lower bounds for the codimension of isometric minimal immersions between round spheres for a given domain dimension and degree the moduli space of such immersions is a compact convex body in a representation space for a lie group of isometries acting transitively on the domain infinitesimal isometric deformations of these minimal immersions give rise to a linear contraction on the moduli and the eigenvalues of the contraction are related to the casimir eigenvalues on the irreducible components of the ambient representation space the study of these eigenvalues leads to new sharp lower bounds for the codimension of the immersions and gives an insight to the subtleties of the boundary of the moduli

symmetric spaces and strongly isotropy irreducible spaces a strongly isotropy irreducible homogeneous space is a connected effective homogeneous space l h where h is a compact subgroup of a lie group l such that the identity component h 0 of h acts irreducibly on t h l h if we assume instead that h but not necessarily h 0 acts irreducibly on tetc l h then l h is called an isotropy irreducible homogeneous space these spaces are interesting because by schur s lemma they all admit an l invariant einstein metric and because they include the irreducible riemannian symmetric spaces as a sub family the nonsymmetric strongly isotropy irreducible homogeneous spaces were classified independently by manturov m1 2 3 wolf wol and later by kr mer kr with some minor omissions in the first two references in addition in wol a detailed study of the geometric properties of strongly isotropy irreducible spaces was carried out the isotropy irreducible spaces which are not strongly isotropy irreducible have recently been classified in wz3 in a different direction we present in this paper a direct proof of a beautiful observation of wall concerning a correspondence between the compact simply connected irreducible symmetric spaces on the one hand and the compact strongly isotropy irreducible quotients of the classical groups on the other hand cf the remarks added in proof in wol pp 147 148 this correspondence has a number of exceptions certain grassmannians and the isotropy irreducible space so 7 g 2 which is diffeomorphic to pt these exceptions appear at first sight to be even more mysterious than the correspondence but our main result is that there is an explanation using general principles of this correspondence as well as all its exceptions in particular this means that the classification of the non symmetric strongly isotropy irreducible quotients of the classical groups may be read off from that of the irreducible symmetric spaces consider a compact simply connected irreducible symmetric space g k where g is the identity component of the isometry group and k is the isotropy group

superpotentials and the cohomogeneity one einstein equations using ideas from convex geometry we prove a classification theorem under suitable hypotheses for superpotentials of the hamiltonian form of the cohomogeneity one ricci flat equations some new superpotentials are also constructed and their associated first order systems analysed the situation with a cosmological constant is briefly examined

a homotopy method for eigenvalue assignment using decentralized state feedback this paper addresses the following problem given an interconnected system m composed of n subsystems of the form a i b i k i i 1 n a i b i a controllable pair and where the off diagonal blocks of m lie in the image of the appropriate b i then is it possible to arbitrarily assign the characteristic polynomial of m by a suitable selection of the characteristic polynomials of a i b i k i moreover is it possible to compute the appropriate characteristic polynomials of the a i b i k i or equivalently construct the k i needed to do so the first question is answered by constructing a mapping f r n rightarrow r n which maps a prescribed set of n of the feedback gains elements of k i i 1 n to the n coefficients of the characteristic polynomial of m the question then becomes given a p in r n does f x p have a solution the answer is found by constructing a homotopy h r n x o 1 rightarrow r n where h x 1 f x and h x 0 is some trivial function degree theory is then applied to guarantee that there exists an x t such that h x t t p for all t in 0 1 the parameterized sard s theorem is then utilized to prove that with probability 1 x t is a smooth curve and hence can be followed numerically from x 0 to x 1 by the solution of a differential equation davidenko s method

on invariance of the euler class groups under a subintegral base change abstract let r be a commutative noetherian ring and r s be a subintegral extension it is proved that the top euler class group of r is isomorphic to the top euler class group of s an example is given to show that similar conclusion does not hold for integral but not subintegral extension of rings

parameter insensitive pole assignment for scalar input systems under the assumption that the parameters affect the systems linearly arbitrary partial insensitive pole placement problem for scalar input systems have been solved a new definition for partial pole placement which takes into account the effect of the unknown varying parameters in the system has been used the result is first given for a system with one parameter and then it is extended to systems with more than one parameter the dual of the same result is also given the paper also emphasizes the structural properties of systems with varying parameters

eigenstructure assignment in linear multivariable systems a parametric solution this paper generalizes a recently reported method 1 of closed loop eigenstructure assignment via state feedback in a linear multivariable system with n states and r control inputs by introducing a lemma on the differentiation of determinants the class of assignable eigenvectors and generalized eigenvectors associated with the assigned eigenvalues is explicitly described by a complete set of n r dimensional free parameter vectors this parametric characterization conveniently organizes the nonuniqueness of the solution of the eigenvalue assignment problem and thereby provides an efficient means of further modifying the system dynamic response a numerical example is worked out to demonstrate the feasibility of the method

orbits of su 2 representations and minimal isometric immersions in contrast to all known examples we show that in the case of minimal isometric immersions of s 3 into s n the smallest target dimension is almost never achieved by an su 2 equivariant immersion we also give new criteria for linear rigidity of a fixed minimal isometric immersion of s 3 into s n the minimal isometric immersions arising from irreducible su 2 representations are linearly rigid within the moduli space of su 2 equivariant immersions hence the question arose whether they are still linearly rigid within the full moduli space we show that this is false by using our new criteria to construct an explicit su 2 equivariant immersion which is not linearly rigid various authors gt to3 w1 have shown that minimal isometric immersions of higher isotropy order d ge 1 play an important role in the study of the moduli space of all minimal isometric immersions of s 3 into s n using a new necessary and sufficient condition for immersions of isotropy order d ge 1 we derive a general existence theorem of such immersions

efficient generation of zero dimensional ideals in polynomial rings let a be a commuta t ive noether ian ring and i an ideal in a it is a wel l known fact that it i 2 2 see 5 when n 1 and r is a regular local ring such as a power series r ing over a field or a regular spot over an infinite perfect field bha twadekar has shown that it ill 2 i for all maximal i d e a l s i n r t see 2 in this paper we generalise these results for the case when i is a zero d imens iona l ideal of the fol lowing kind i fo r any zero d imensional ideal i in a r t 1 t we have it i i2 it i if n 2 2 let i b e a zero d imensional ideal in a r t then it i i2 i in each o f the following cases a r is a power series ring over a field b r is a regular local r ing of an affine a lgebra over an infinite perfect field in sect 2 we state some theorems without proof which we require later in sect 3 we prove the analogues of dav i s ge rami t a theorems in sect 4 we prove the result for a power series r ing or for a regular local r ing of an affine algebra the results of this pape r go th rough for laurent po lynomia l rings

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

a continuation algorithm for eigenvalue assignment by decentralized constant output feedback this paper describes a continuation approach to eigenvalue assignment by decentralized constant output feedback for interconnected systems the method is a homotopy technique which embeds the decentralized control problem into a parametrized family of control problems the parametrization is based on the system interconnection structure and represents a continuous deformation of a system consisting of the uncoupled subsystems into the original system with full interconnections based on such a deformation a differential equation is constructed whose solution trajectory has an endpoint which is a decentralized constant output feedback matrix assigning the desired spectrum to the interconnected system the derivation of the differential equation and considerations which guarantee the existence of a solution are given two examples are presented

stable range one for rings with many idempotents an associative ring r is said to have stable range 1 if for any a b e r satisfying ar br r there exists y e r such that a by is a unit the purpose of this note is to prove the following facts theorem 3 an exchange ring r has stable range 1 if and only if every regular element of r is unit regular theorem 5 if r is a strongly 7r regular ring with the property that all powers of every regular element are regular then r has stable range 1 the latter generalizes a recent result of goodearl and menal 5 let r be an associative ring with identity r is said to have stable range 1 if for any a b e r satisfying ar br r there exists y e r such that a by is a unit this definition is left right symmetric by vaserstein 9 theorem 2 furthermore by a theorem of kaplansky all one sided units are two sided in rings having stable range 1 cf vaserstein 10 theorem 2 6 it is well known that a von neumann regular ring r has stable range 1 if and only if r is unit regular see for example goodearl 4 proposition 4 12 call a ring r strongly 7 regular if for every element a e r there exist a number n depending on a and an element x e r such that an an lx this is in fact a two sided condition 3 it is an open question whether all strongly 7r regular rings have stable range 1 goodearl and menal 5 proved that strongly 7r regular rings are unit regular and hence have stable range 1 theorem 5 8 p 278 in this note we first extend the above result for von neumann regular rings to a larger class of rings which includes all strongly 7r regular rings 7r regular rings von neumann regular rings and algebraic algebras as an application of this we prove that a strongly 7r regular ring r has stable range 1 if powers of every regular element are regular the latter is a generalization of the abovementioned result of goodearl and menal for strongly 7r regular regular rings as one can see from our proofs rings in these classes have a large supply of idempotents throughout r stands for an associative ring with identity and j r for the jacobson radical of r modules are unitary right r modules except otherwise specified for other undefined terms readers are referred to 4 let mr be a right r module following crawley and jonsson 2 mr is said to have the exchange property if for every module ar and any two received by the editors january 24 1994 and in revised form may 3 1994 originally communicated to the proceedings of the ams by k a goodearl 1991 mathematics subject classification primary 1 6d70 1 6p70

the euler class group of a polynomial algebra with coefficients in a line bundle let a be a commutative noetherian ring and p be a projective a module of rank text dim a 1 an intriguing open question is to find the precise obstruction for p to split as p simeq q oplus a for some a module q in this paper we settle this question when a r t for some ring r containing the field of rationals and p is a projective a module of rank text dim r

feedback gain optimization in decentralized eigenvalue assignment abstract this paper develops a new design procedure for minimizing the norm of a decentralized output feedback matrix which assigns a user specified set of eigenvalues two distinct approaches which can be meshed together as a unified design tool are derived assuming one has computed a decentralized feedback matrix which assigns a desired spectrum the first approach describes an iterative algorithm which reduces an algebraic cost function e g the frobenius norm of the feedback gains while maintaining the desired spectrum this algorithm allows for small movements in the eigenvalues the iteration step is based on the first order variational behaviour of the eigenvalue eigenvector equations the second algorithm modifies a continuation method for decentralized eigenvalue assignment to include an optimizing factor numerical considerations for the design procedures are discussed an example showing the improvement possible by the application of the procedure is also given

a note on complete intersections let r be a regular local ring and let r t be a polynomial algebra in one variable over r in this paper the author proves that every maximal ideal of r t is complete intersection in each of the following cases 1 r is a local ring of an affine algebra over an infinite perfect field 2 r is a power series ring over a field introduction let f be a regular local ring let r t be a polynomial algebra in one variable over r in d g the following question has been asked question is every maximal ideal of r t complete intersection in this paper we prove that the answer to the above question is affirmative in each of the following cases 1 f is a local ring of an affine algebra over an infinite perfect field 2 f is a power series ring over a field this paper is divided into three sections in 1 we fix notations and state a theorem without proof which is used in 2 and 3 in 2 we prove some lemmas and propositions which are used in proving the result when f is a local ring of an affine algebra 3 deals with the power series case 1 throughout this paper we consider commutative noetherian rings with 1 for a ring r dim r denotes its krull dimension which we always assume to be finite if r is a local ring then 911 f will always denote its unique maximal ideal if m is a finitely generated f module then ii m will denote the minimal number of genera tors of m for an ideal i of r i i denotes the height of definition let be an unmixed ideal of r of height r then i is said to be complete intersection in r if i 2lri xrai where ax a2 ar is a regular r sequence remark if r is cohen macaulay then i is complete intersection if and only if p ht 7 let r and s be two local rings definition r is said to be a local extension of s if s is a subring of r and 91t s 911 f d s r is said to be unramified over s if gjt s r 911 f and f 91t f is separable over 5 91l s let l k be a finite separable extension of k then l is a simple extension of k by a minimal polynomial of l over k we always mean an irreducible monic polynomial over k satisfied by a generator of l over k now we state a theorem which has been proved in d g theorem 3

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

the milnor patching property abstract this paper gives a complete answer to the following question if r is a commutative ring in which every height 1 ideal is a local set theoretic complete intersection e g r could be regular and r 1 and r 2 are r algebras which are flat or of finite type when is it the case that projective r modules are precisely those r modules constructed by patching a projective r 1 module to a projective r 2 module over r 1 r r 2 in other words we give a classification theorem for situtations in which the conclusion to the milnor patching theorem holds our answer takes the form of a necessary and sufficient condition whose truth or falsity is very easy to check in most particular situations

a schur method for pole assignment in this paper a new numerical algorithm for pole assignment of linear time invariant systems is presented the proposed algorithm uses reliable numerical techniques based on the schur form of the state matrix and on the use of orthogonal transformations only the bad eigenvalues of the system are modified by a sequential pole shifting procedure and the resulted gains in the feedback matrix are minimized

on maximal ideals in polynomial and laurent polynomial rings abstract when r is a complete local ring containing a field of characteristic zero we show that any maximal ideal in r t t 1 is projectively generated if r is a regular ring containing a field of characteristic zero we show that any maximal ideal m of r t such that ht m dim r is efficiently generated

linearity in the closed loop characteristic polynomial this paper examines when the coefficients of the closed loop characteristic polynomial are linear in the elements of the feedback matrix an error in previous work tarokh 1980 is pointed out necessary and sufficient conditions are given for all of the coefficients to be linear in the elements of the output feedback matrix

successive pole shifting by state controllers of prescribed structure given a linear time invariant multivariable system a design procedure is developed representing a straightforward approach to the problem of constructing an appropriate feedback matrix of prescribed structure by successive shifting of selected system poles involving the concept of the moore penrose psoudoin verse a solution of the linearized definitive equations of the pole assignment problem can be attained which furthermore tends to favour small feedback gains the suggested method is simple in theory direct in application and without ticklish steps a simple example is included to illustrate the idea and its implications to demonstrate its applicability and practical usefulness an incomplete state feedback is designed for the example of an 11th order system

an algorithm for pole assignment of time invariant linear systems an algorithm is suggested for the computation of a linear state feedback for a single input system such that the resultant closed loop system has specified eigenvalues the algorithm is more efficient than many comparable techniques and has some desirable numerical properties it is closely related to the qr algorithm for the eigenproblem and extends to descriptor systems