Some generic invariant factor assignment results using dynamic output feedback

Working with input-output transfer functions in the frequency domain and exploiting a formulation involving generalized Sylvester resultants, we are able to derive necessary and sufficient conditions for generic invariant factor assignment, in several cases, using proper dynamic output feedback compensators.     

A generalization of right pci rings

By a well-known result of Osofsky [6, Theorem] a ring R is semisimple (i.e. R is right artinian and the Jacobson radical of R is zero) if and only if every cyclic right R-module is injective. Starting from this, a larger class of rings has been introduced and investigated, namely the class of right PCI rings. A ring R is called right PCI if every proper cyclic right R- module is injective (proper here means not being isomorphic to RR). By [l] and [Z], a right PCI ring is either semisimple or it is a right noetherian, right hereditary simple ring. The latter ring is usually called a right PCI domain. In this paper we consider the similar question in studying rings whose cyclic right modules satisfy some decomposition property. The starting point is a theorem recently proved in 13, Theorem 1.1): A ring R is right artinian if and only if every cyclic right R- module is a direct sum of an injective module and a finitely cogenerated module.     

New algorithm to construct a stabilizing controller for linear systems over integral domains

We are concerned with the coordinate-free approach to controlsystems, which is a factorization approach but does not requirethe coprime factorizations of plants, provided only that theset of stable causal transfer functions forms an integral domain.We first present a new criterion of the feedback stabilizability.This is a generalization of the criteria in terms of reducedminors given by Sule in 1994 and in terms of quotient idealsgiven by Shankar and Sule in 1992. We then present an algorithmto construct a stabilizing controller, which is derived fromthe result given by Lin in 1998. Sule's algorithm and its derivationsneed to convert some matrices over some ring of fractions toones over the set of stable causal transfer functions, whichmay raise the unnecessarily complicated calculations. On theother hand, the algorithm of this paper does not require sucha conversion. An illustrative example to construct a stabilizingcontroller is also presented.     

关于对偶模的平坦性和内射性(Ⅱ)

对交换环Ｒ和Ｂ－模范畴上的一个内射余生成元Ｂ,我们用相对于Ｅ的对偶模的性质刻画了ＱＦ环,ＩＦ环和半遗传环．     

On certain homological finiteness conditions

In this paper, we show that the injective dimension of all projective modules over a countable ring is bounded by the self-injective dimension of the ring. We also examine the extent to which the flat length of all injective modules is bounded by the flat length of an injective cogenerator. To that end, we study the relation between these finiteness conditions on the ring and certain properties of the (strict) Mittag–Le?er modules. We also examine the relation between the self-injective dimension of the integral group ring of a group and Ikenaga’s generalized (co-)homological dimension.     

An algebraic analysis approach to linear time-varying systems

^** Email: zerz{at}mathematik.uni-kl.de This paper introduces an algebraic analysis approach to lineartime-varying systems. The analysis is carried out in an ‘almosteverywhere’ setting, i.e. the considered signals are smoothexcept for a set of measure zero, and the coefficients of thelinear ordinary differential equations are supposed to be rationalor meromorphic functions. The methodology is based on a normalform for matrices over the resulting ring of differential operators,which is a non-commutative analogue of the Smith form. Thisis used to establish a duality between linear time-varying systemson the one hand and modules over the ring of differential operatorson the other. This correspondence is based on the fact thatthe signal space is an injective cogenerator when consideredas a module over this ring of differential operators.     

Rings with finite essential socle

Let be a ring such that every direct summand of the injective envelope has an essential finitely generated projective submodule. We show that, if the cardinal of the set of isomorphism classes of simple right -modules is no larger than that of the isomorphism classes of minimal right ideals, then cogenerates the simple right -modules and has finite essential socle. This extends Osofsky's theorem which asserts that a right injective cogenerator ring has finite essential right socle. It follows from our result that if is a CS cogenerator, then is already an injective cogenerator and, more generally, that if is CS and cogenerates the simple right -modules, then it has finite essential socle. We show with an example that in the latter case need not be an injective cogenerator.

     

Current algebras, highest weight categories and quivers

We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametrized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on the affine weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to an interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and star-shaped type, as well as tame quasi-hereditary algebras, that arise from our study.     

Convexity and Weighted Integral Inequalities for Energy Decay Rates of Nonlinear Dissipative Hyperbolic Systems

This work is concerned with the stabilization of hyperbolic systems by a nonlinear feedback which can be localized on a part of the boundary or locally distributed. We show that general weighted integral inequalities together with convexity arguments allow us to produce a general semi-explicit formula which leads to decay rates of the energy in terms of the behavior of the nonlinear feedback close to the origin. This formula allows us to unify for instance the cases where the feedback has a polynomial growth at the origin, with the cases where it goes exponentially fast to zero at the origin. We also give three other significant examples of nonpolynomial growth at the origin. Our work completes the work of [15] and improves the results of [21] and [22] (see also [23] and [10]). We also prove the optimality of our results for the one-dimensional wave equation with nonlinear boundary dissipation. The key property for obtaining our general energy decay formula is the understanding between convexity properties of an explicit function connected to the feedback and the dissipation of energy.     

Injective hulls are not natural

In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are injective. In particular, assigning to a field its algebraic closure, to a poset or Boolean algebra its Mac-Neille completion, and to an R-module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation. Received January 21, 2000; accepted in final form August 10, 2001. RID="h1" RID="h2" RID="h3" ID="h1"The hospitality of York University is gratefully acknowledged by the first author. ID="h2"Third author partially supported by the Grant Agency of the Czech Republic under Grant no. 201/99/0310, and the hospitality of York University is also acknowledged. ID="h3"Partial financial assistance by the Natural Sciences and Engineering Councel of Canada is acknowledged by the fourth author.     

A class of congruences on a posemigroup

V-completions of partially ordered algebraic systems have been studied by many authors, see Jaffard [7]. In particular, Burgess and McFadden [3] considered V-completions of posemigroups by means of ideal systems. In seemingly unrelated work, McFadden [10] characterized certain what we shall call ‘strongly isotone’ m-homomorphisms on posemi-groups; while Bigard [1], Dubreil-Jacotin [6] and McFadden [11] studied group homomorphisms on posemigroups. By determining certain coreflections [8], we prove that any surjective isotone homomorphism on a posemigroup can be factored into an injective strongly isotone homomorphism followed by a surjective strongly isotone m-homorphism. This result is used to bring the previous theories together and to generalise them.     

INJECTIVE PRECOVERS AND MODULES OF GENERALIZED INVERSE POLYNOMIALS

Let (S,≤) be an ordered set. Recall that (S,≤) is artinian if every strictly decreasingsequence of elements of S is ?nite, and that (S,≤) is narrow if every subset of pairwiseorder-incomparable elements of S is ?nite. Let S be a commutative monoid. Unl…     

Exactness of direct limits for abelian categories with an injective cogenerator

We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg–Moore category of a monad on the category of sets preserve regular epimorphisms.     

Rings characterized by injectivity classes

Recently, there are a number of well known theorems which char­acterize rings in terms of injective modules (cf. [2], [3], [5] and [8]). The purpose of this paper is to give general theorems which show that some similar results remain true when many other classes of general injective modules are considered in place of injective modules. Our results encompass several well known results by Huynh and Smith, Liu, and Xue. In particular we answer negatively the question of Yue Chi Ming [11] and an example is given to show that E—injective modules form an intermediate class between continuous and quasi-continuous modules(cf. Example 3).     

非线性最优控制的反馈逼近

本文将在线性最优控制中经典的Ｋｌｅｉｎｍａｎ－Ｎｅｗｔｏｎ［１］方法推广到非线性的最优控制问题。我们证明,由一个稳定反馈控制生成的反馈控制序列在原点附近会一致逼近最优控制,并指出这一研究在Ｈ∞－非线性最优控制中的作用。     

Testing sets for properness of polynomial mappings

In this paper we characterize testing sets for properness of polynomial mappings . We also study the set of points at which such mappings are not proper. As the first application we give a proof of the Complementary Conjecture of McKay and Wang (Conjecture 12 in [16]). The second application is an answer to the problem of Kraft- Russell about a characterization of The third application is (given in [13]) a solution of the Problem of Van den Essen and Shpilrain about endomorphism of the ring The fourth application is a theorem about extensions of affine varieties. Received: 27 February 1998 / in revised form: 10 January 1999     

Two-dimensional regularity and exactness

We define notions of regularity and (Barr-)exactness for 2-categories. In fact, we define three notions of regularity and exactness, each based on one of the three canonical ways of factorising a functor in Cat: as (surjective on objects, injective on objects and fully faithful), as (bijective on objects, fully faithful), and as (bijective on objects and full, faithful). The correctness of our notions is justified using the theory of lex colimits [12] introduced by Lack and the second author. Along the way, we develop an abstract theory of regularity and exactness relative to a kernel–quotient factorisation, extending earlier work of Street and others  and .     

Completion of Modules over Serial, Right Noetherian Rings

Linear topology defined on an arbitrary right module over a right Noetherian serial ring R enables one to describe the reduced, pure injective R-modules as modules that are complete in this topology. With the use of the completion of modules, the pure injective envelope of any right R-module is constructed. Bibliography: 8 titles.     

Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

Let be a finite-dimensionalrepresentation of a connected reductive complex Lie group . Denote by G the derived subgroup of and assume that the categorical quotient is one dimensional, i.e. for a nonconstant polynomial f. In this situationthere exists a homomorphism , the radial component map, where is the first Weyl algebra. We show that theimage of is isomorphicto the spherical subalgebra of a rational Cherednik algebrawhose multiplicity function is defined by the roots of the Bernstein–Satopolynomial of f. In the case where is also multiplicity free we describe the kernel of and prove a Howe duality result between representationsof G occurring in andlowest-weight modules over the Lie algebra generated by f andthe "dual" differential operator ; this extends results of H. Rubenthaler obtained when is a parabolic prehomogeneous vector space.If satisfies a Capelli-typecondition, some applications are given to holonomic and equivariantD-modules on V. These applications are related to results provedby Muro [34, 37, 38] or Nang [40, 42, 44] in special cases ofthe representation .