a structure theorem for noetherian p i rings with global dimension two abstract let a be the artin radical of a noetherian ring r of global dimension two we show that a rer where e is an idempotent a contains a heredity chain of ideals and the global dimensions of the rings r a and ere cannot exceed two assume further than r is a polynomial identity ring let p be a minimal prime ideal of r then p p 2 and the global dimension of r p is also bounded by two in particular if the krull dimension of r p equals two for all minimal primes p then r is a semiprime ring in general every clique of prime ideals in r is finite and in the affine case r is a finite module over a commutative affine subring additionally when a 0 the ring r has an artinian quotient ring and we provide a structure theorem which shows that r is obtained by a certain subidealizing process carried out on rings involving dedekind prime rings and other homologically homogeneous rings

quantum function algebra at roots of 1 abstract we introduce a form of the quantum function algebra on a drinfeld jimbo quantum group over the ring z q q 1 specializing q to a root of 1 we show that over the cyclotomic field this algebra is a projective module over its central sub algebra which is the usual coordinate algebra of the group we study the induced poisson lie structure of the group a bundle of algebras on a complex simply connected lie group with hamiltonian flows in the bundle is constructed some representations of the quantum function algebra in a root of 1 are constructed as an application an estimate of the dimension of an arbitrary representation is given

additive mappings decreasing rank one abstract let b x be the algebra of bounded operators on a real or complex banach space x and f x a subalgebra of finite rank operators a complete description of additive mappings f x f x which map rank one operators to operators of rank at most one is given

ring isomorphisms and linear or additive maps preserving zero products on nest algebras it is shown that every ring isomorphism between norm closed standard subalgebras of nest algebras acting on infinite dimensional banach spaces is either a spatial isomorphism or a spatial conjugate isomorphism this is then used to characterize the linear or additive maps between nest algebras which preserve zero products

additive preservers of rank additivity on matrix spaces abstract let f be a field let v denote the vector space of all m n matrices over f or the vector space of all n n symmetric matrices over f of characteristic not 2 or 3 for each fixed positive integer s 2 let q s denote the set of all matrix pairs a b in v such that rank a b rank a rank b s we characterize additive mappings on v such that a b q s whenever a b q s for a fixed s we also describe the structure of linear mappings from the space of n n matrices over f to the space of p q matrices over f that preserve rank additivity where char f 2

azumaya loci and discriminant ideals of pi algebras abstract we prove that under mild assumptions for all positive integers l the zero set of the discriminant ideal d l r z r tr of a prime affine polynomial identity pi algebra r coincides with the zero set of the modified discriminant ideal m d l r z r tr of r and give an explicit description of this set in terms of the dimensions of the irreducible representations of r furthermore we prove that when l is the square of the pi degree of r this zero set is precisely the complement of the azumaya locus of r this description is used to classify the azumaya loci of the multiparameter quantized weyl algebras at roots of unity as another application we prove that the zero set of the top discriminant ideal of a prime affine pi algebra r coincides with the singular locus of the center of r provided that the discriminant ideal has height at least 2 r has finite global dimension and r is a cohen macaulay module over its center

central singularities of quantum spaces abstract let a be a positively graded connected affine c algebra generated in degree one which is a finite module over a central subring r let a be the structure sheaf over proj r and spec z the central proj if injdim a spec z has singularities unless the ramification locus of a is pure of codimension one if gldim a a and the singular locus of spec z coincide

sugawara operators and kac kazhdan conjecture soit g une algebre de lie de kac moody affine de type x l 1 soit l le module de poids le plus eleve unique irreductible pour chaque poids de kac kazhdan on a chl exp re 1 exp 1

additive rank one preservers on block triangular matrix spaces in this note we address a description of the general form of additive rank one preservers on block triangular matrix spaces over an arbitrary field as an application we characterize additive mappings preserving rank additivity on block triangular matrix algebras

a feedback approach to the steady state performance of fractionally spaced blind adaptive equalizers this paper proposes a new approach to the analysis of the steady state performance of constant modulus algorithms cma which are among the most popular adaptive schemes for blind equalization a major feature of the proposed feedback approach is that it bypasses the need for working directly with the weight error covariance matrix in so doing approximate expressions for the steady state mean square error of several cm algorithms are derived including cma2 2 cma1 2 normalized cma and a new normalized cma variant with less bias a comparison among the various algorithms is also performed along with several simulation results the conclusions confirm the superior performance of cma2 2

polynomial identity rings as rings of functions ii abstract in characteristic zero zinovy reichstein and the author generalized the usual relationship between irreducible zariski closed subsets of the affine space their defining ideals coordinate rings and function fields to a non commutative setting where varieties carry a pgl n action regular and rational functions on them are matrix valued coordinate rings are prime polynomial identity algebras and function fields are central simple algebras of degree n in the present paper much of this is extended to prime characteristic in addition a mistake in the earlier paper is corrected one of the results is that the finitely generated prime pi algebras of degree n are precisely the rings that arise as coordinate rings of n varieties in this setting for n 1 the definitions and results reduce to those of classical affine algebraic geometry

convergence analysis of the cma blind equalizer the convergence behaviour of the constant modulus algorithm cma is analytically evaluated in terms of the mean and the variance of the equalized sequence conditioned on the transmitted symbol thus characterizing the dispersion pattern of the equalized samples around the qam constellation points this extends to the cma and improves the approach of weerackody et al see ibid vol 40 no 6 p 1383 1992 where the mse trajectory of the generalized sato algorithm gsa was analysed

convergence analysis of self adaptive equalizers a theoretical analysis of self adaptive equalization for data transmission is carried out starting from known convergence results for the corresponding trained adaptive filter the development relies on a suitable ergodicity model for the sequence of observations at the output of the transmission channel thanks to the boundedness of the decision function used for data recovery it can be proved that the algorithm is bounded strong convergence results can be reached when a perfect noiseless equalizer exists the algorithm will converge to it if the eye pattern is initially open otherwise convergence may take place towards certain other stationary points of the algorithm for which domains of attraction have been defined some of them will result in a poor error rate the case of a noisy channel exhibits limit points for the algorithm that differ from those of the classical trained algorithm the stronger the noise the greater the difference is one of the principal results of this study is the proof of the stability of the usual decision feedback algorithms once the learning period is over

on two sided modules which are left projective abstract the purpose of this paper is to study an analogue for non commutative rings r of projective modules we consider r bimodules of the form m v p where v rn is the module of row vectors and p is any two sided submodule of v these are finitely generated central r bimodules our main results 1 11 1 12 are these let ik denote the ideal generated by the k k minors of matrices with rows in p then p is projective and of rank d as left module if in d r and in d 1 0 the converse holds if every primitive ideal of r is maximal thus the familiar criterion of commutative algebra carries over directly to this class of rings and left projectivity is equivalent with right projectivity for central bimodules 1 13 some related questions are discussed in sections 2 and 3

on the convergence of blind adaptive equalizers for constant modulus signals this paper studies the behavior of the error sequence of stop anti go variants of two adaptive blind equalizers namely cma2 2 and sato s 1975 algorithm it is shown that for transmitted signals with constant modulus spl gamma the equalizer output can be made to lie within the circle of radius spl gamma spl radic c infinitely often for some value of c that is only slightly larger than one

additive preservers of non zero decomposable tensors let t be an additive mapping from a tensor product of vector spaces over a field into itself we describe t for the following two cases i t is surjective and sends non zero decomposable elements to non zero decomposable elements and ii t a is a non zero decomposable element if and only if a is a non zero decomposable element

convergence and performance analysis of godard family and multimodulus algorithms for blind equalization we obtain the convergence of the godard family including the sato and constant modulus cm algorithms and the multimodulus algorithms mma in a unified way our analysis also works for cma fractionally spaced equalizer fse our assumptions are quite realistic the channel input can be asymptotically stationary and ergodic the channel impulse response is finite and can be stationary and ergodic this models fading channels and the equalizer length is finite the noise is independent and identically distributed i i d the channel input can be discrete or continuous our approach allows us to approximate the whole trajectory of the equalizer coefficients this provides estimates of the rate of convergence and the system performance symbol error rate can be evaluated under transience and steady state

relationships between the constant modulus and wiener receivers the godard 1980 or the constant modulus algorithm cma is an effective technique for blind receiver design in communications however due to the complexity of the constant modulus cm cost function the performance of the cm receivers has primarily been evaluated using simulations theoretical analysis is typically based on either the noiseless case or approximations of the cost function the following question while resolvable numerically for a specific example remains unanswered in a generic manner in the presence of channel noise where are the cm local minima and what are their mean squared errors mse in this paper a geometrical approach is presented that relates the cm to wiener or minimum mse receivers given the mse and the intersymbol user interference of a wiener receiver a sufficient condition is given for the existence of a cm local minimum in the neighborhood of the wiener receiver the mse bounds on cm receiver performance are derived and shown to be tight in simulations the analysis shows that while in some cases the cm receiver performs almost as well as the nonblind wiener receiver it is also possible that due to its blind nature the cm receiver may perform considerably worse than a nonblind wiener receiver

bounds for the mse performance of constant modulus estimators the constant modulus cm criterion has become popular in the design of blind linear estimators of sub gaussian independent and identically distributed i i d processes transmitted through unknown linear channels in the presence of unknown additive interference in this paper we present an upper bound for the conditionally unbiased mean squared error umse of cm minimizing estimators that depends only on the source kurtoses and the umse of wiener estimators further analysis reveals that the extra umse of cm estimators can be upper bounded by approximately the square of the wiener i e minimum umse since our results hold for vector valued finite impulse response infinite impulse response fir iir linear channels vector valued fir iir estimators with a possibly constrained number of adjustable parameters and multiple interferers with arbitrary distribution they confirm the longstanding conjecture regarding the general mean square error mse robustness of cm estimators

homological integral of hopf algebras the left and right homological integrals are introduced for a large class of infinite dimensional hopf algebras using the homological integrals we prove a version of maschke s theorem for infinite dimensional hopf algebras the generalization of maschke s theorem and homological integrals are the keys to studying noetherian regular hopf algebras of gelfand kirillov dimension one

classical localizability in solvable enveloping algebras and poincare birkhoff witt extensions abstract in several classes of noncommutative noetherian rings jategaonkar s intersection conditions are verified for finite unions of cliques of prime ideals and it follows that these finite unions of cliques are classically localizable if they satisfy incomparability among the rings in question are the enveloping algebra of any finite dimensional solvable lie algebra g over any field k of characteristic zero any twisted smash product r u g where r is a commutative noetherian k algebra and g acts on r via k linear derivations and certain iterated differential operator rings over r in contrast to previously verified cases k is allowed to be countable one tool used here a lying over theorem for cliques has some independent interest namely if r s is a flat finite centralizing extension of noetherian rings satisfying the second layer condition and if x is a clique in spec r there exists a clique y in spec s such that p r p y x

on 2 local automorphisms and 2 local isometries of b h abstract it is an important result of semrl which states that every 2 local automorphism of the full operator algebra over a separable hilbert space is necessarily an automorphism in this paper we strengthen that result quite substantially for automorphisms indeed we show that one can compress the defining two equations of 2 local automorphisms into one single equation hence weakening the requirement significantly but still keeping essentially the conclusion that such maps are necessarily automorphisms

integrality for pi rings abstract in this paper we study schelter integral extensions of pi rings we prove in particular lying over going up and incomparability for prime ideals a major result is transitivity of integrality if r s b are pi rings such that b is integral over s and s is integral over r then b is integral over r next we obtain a powerful criterion for integrality if s is a prime pi ring such that its center is integral over a noetherian subring r of s then s is integral over r this allows interesting applications to the theory of finite group actions further topics concern eakin nagata type results and embeddings of quotient rings for integral extensions finally we analyze the relationship between module finite extensions and finitely generated integral extensions obtaining positive results for affine noetherian pi algebras and algebras satisfying certain restrictions on pi degrees e g algebras of low pi degree

additive decomposability preservers and related results on tensor products of matrices abstract in this paper we characterize additive maps between tensor spaces that send decomposable tensors to decomposable tensors as an application we classify all additive maps from tensor products of spaces of rectangular matrices to spaces of rectangular matrices which do not increase the rank of tensor product of rank one matrices

convergence analysis of an algorithm for blind equalization the authors consider a key algorithm for blind equalization and derive expressions for the evolution of the equalizer output error trajectory the authors develop a model to examine the convergence behavior of this algorithm suitable approximations are incorporated into the model to facilitate analysis the validity of these approximations is demonstrated for a typical communication channel it is shown that even for a channel for which the assumption of gaussianity of the equalizer input data may not be very good the analysis presented predicts the convergence behavior reasonably well

properties of some blind equalization criteria in noisy multiuser environments blind equalization in noisy multiuser channels has met with increasing attention with the advent of multiaccess digital communication systems we examine blind equalizer performance in cases where perfect equalization proves unattainable due to noise and interference from concurrent users in particular we obtain a characterization of stationary points and extrema for a family of blind criteria in undermodeled cases which assimilates the influence of differing source statistics and background noise correlation properties relations to mean square equalization measures are then obtained as a byproduct by re examining a gradient search procedure we obtain domains of attraction of each extremum in a special sufficient order setting we also derive a global step size bound for undermodeled cases which ensures convergence of a gradient search procedure to an extremum of a blind cost function we likewise confirm that the super exponential algorithm results from an optimal choice of this step size parameter

pbw deformation theory and regular central extensions a deformation u of a graded k algebra a is said to be of pbw type if gru is a it has been shown for koszul and n koszul algebras that the deformation is pbw if and only if the relations of u satisfy a jacobi type condition in particular for these algebras the determination of the pbw property is a finite and explicitly determined linear algebra problem we extend these results to an arbitrary graded k algebra using the notion of central extensions of algebras and a homological constant attached to a which we call the complexity of a

linear additive preservers of rank 2 on spaces of alternate matrices over fields abstract let kn f be the linear space of all n n alternate matrices over a field f and let k n 2 f be its subset consisting of all rank 2 matrices an operator kn f kn f is said to be additive if a b a b for any a b kn f linear if is additive and aa af a for every a f and a kn f and a preserver of rank 2 on kn f if k n 2 f k n 2 f when n 4 we characterize all linear respectively additive preservers of rank 2 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself

a note on additive mappings decreasing rank one kuzma characterized additive mappings on the space of all finite rank bounded linear operators on a real or complex banach space that decreases operators of rank one in this note we give a short proof of his result in a slightly more general setting of the tensor product of a right and a left vector space over a division ring

integral ring homomorphisms abstract the purpose of this paper is to extend the classical notion of integral extensions of commutative rings to homomorphisms of affine pi rings it generalizes previous work r pare and w schelter j algebra 53 1978 477 479 w schelter j algebra 40 1976 245 254 errata 44 1979 576 which assumed the existence of centralizing elements without such elements the close relation between integrality and module finiteness breaks down cf examples 1 3 1 4 but the geometric implications of integrality remain our main result theorem 6 3 is an analogue of chevalley s theorem that a homomorphism r s is integral if and only if the induced map on spectra is proper it is proved in sections 6 8 by combining a geometric analysis with some explicit estimates of degrees a corollary is that the composition of integral homomorphisms is integral proposition 6 10 the main new tools we use are central homomorphisms of a ring r to orders over dedekind domains which we call curves we prove theorem 5 13 that the boundary of any constructible set in x spec r contains a dense set of points accessible along such a curve this fact contains a description corollary 5 14 of the zariski topology on x as well as the theorem of bergman and small on degrees of representations corollary 5 15 some of the results of this paper were announced in m artin in proceedings international symposium on algebraic geometry kyoto 1977 pp 237 247

homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension abstract we prove that a noetherian hopf algebra of finite global dimension possesses further attractive homological properties at least when it satisfies a polynomial identity this applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity as well as to classical enveloping algebras in positive characteristic in all three cases we show that these algebras are auslander regular and macaulay we derive representation theoretic consequences concerning the coincidence of the non azumaya and singular loci for each of the above three classes of algebras

exact sequences of inner automorphisms of tensors abstract we produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor our sequence generalizes known sequences for associative and non associative algebras in a manner similar to those our sequence facilitates inductive reasoning about and calculation of the groups of symmetries of a tensor the new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics

equalization and carrier phase recovery of cma and mma in blind adaptive receivers this paper derives general formulae for normalized error surface curvatures by using the constant modulus algorithm cma the multimodulus algorithm mma and two well known blind carrier phase recovery algorithms cpra at the stationary points with arbitrary two dimensional symmetric signal constellations based on the derived error surface curvatures both the performance of the cma followed by a blind cpra and the performance of the mma are forecasted and evaluated in terms of blind equalizer convergence and carrier phase recovery

local automorphisms of standard operator algebras let x be an infinite dimensional separable real or complex banach space and a a closed standard operator algebra on x then every local automorphism of a is an automorphism the assumptions of infinite dimensionality separability and closeness are all indispensable

jordan zero product preserving additive maps on operator algebras let a b be an additive surjective map between some operator algebras such that ab ba 0 implies a b b a 0 we show that under some mild conditions is a jordan homomorphism multiplied by a central element such operator algebras include von neumann algebras c algebras and standard operator algebras etc particularly if h and k are infinite dimensional real or complex hilbert spaces and a b h and b b k then there exists a nonzero scalar c and an invertible linear or conjugate linear operator u h k such that either a cuau 1 for all a b h or a cua u 1 for all a b h

a homological reformulation of the link condition abstract we prove an equivalent condition for the existence of a link between prime ideals in terms of the structure of a certain cohomology module we use this formulation to answer an open question regarding the nature of module extensions over one sided noetherian rings we apply the techniques developed in this paper to the local link structure of prime ideals of small homological height and examine when certain noetherian rings satisfy the density condition

effect of channel noise on blind equalization and carrier phase recovery of cma and mma this work investigates the effect of channel noise on the performances of the constant modulus algorithm cma the multimodulus algorithm mma and a phase tracker pt following the cma for carrier phase recovery when various symmetrical two dimensional complex valued source constellations are used the contributions of this work are two fold first a generalized formulation of the relocation of stationary points toward the origin owing to channel noise is derived for both the cma and the mma allowing their sensitivity to channel noise to be explicitly compared when various constellations are used second the curvatures of the error surface at the stationary points in both the radial and the angular directions are reduced in a way that is linearly proportional to the square of the distances between the stationary points and the origin as the stationary points of the cma and the mma relocate toward the origin in the presence of additive channel noise the reduction of the error surface curvature in the radial direction reduces the rate of blind equalization convergence whereas the reduction in the angular direction degrades the capacity for carrier phase recovery

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

endomorphisms of right ideals of the weyl algebra let a a k be the first weyl algebra over an infinite field k let p be any noncyclic projective right ideal of a and set s end p we prove that as k algebras s f a in contrast there exists a noncyclic projective right ideal q of s such that s end q thus despite the fact that they are morita equivalent s and a have surprisingly different properties for example under the canonical maps autk a pick a pick s in contrast autk s has infinite index in pick s introduction given a comnmutative domain r the first weyl algebra a r is defined to be the associative r algebra thus r is central subring generated by elements x and y subject to the relation xy yx 1 when no ambiguity is possible we write a for a r let f be a field of characteristic zero then a a f is a simple ring and indeed may be thought of as one of the nicest and most important examples of simple noetherian rings the initial motivation for this paper was the following result of smith if p is the noncyclic right ideal p x2a xy 1 a of a then end p t a as f algebras 11 note that as char f 0 a is a simple hereditary ring and so end p is automatically morita equivalent to a even worse as pedp ada 17 the full matrix rings m2 a and m2 end p are isomorphic thus any proof of smith s result must be fairly subtle and may therefore provide useful invariants for a the first main aim of this paper is to generalize smith s result to an arbitrary projective right ideal of a while the proof of this is harder than that of smith s result it does provide a more informative proof in the sense that unlike smith we do not and cannot require an explicit description of end p for the rest of this introduction k will denote a field of arbitrary characteristic and all isomorphisms of rings will be k algebra isomorphisms theorem a let p be a projective right ideal of a a k then end p a if and only if p is a cyclic right ideal of a see theorem 3 1 this has some easy corollaries corollary b let p and q be projective right ideals of a a k then end p end q if and only if p to q for some t e d k the division ring of fractions of a and a e autk a the group of k automorphisms of a received by the editors february 13 1986 1980 mathematics subject classification 1985 revision primary 16a65 16a72 16a19 supported in part by an nsf grant 1987 american mathematical society 0002 9947 87 1 00 25 per page

trace rings of generic matrices are cohen macaulay in this paper we prove that trace rings of generic matrics are cohen macaulay theorem 7 3 6 this is done by relating this problem to a conjecture of stanley about modules of invariants under a reductive group we prove a slightly weakened version conjecture 3 4 of this conjecture in special cases theorem 6 1 8 in particular we obtain that conjecture 3 4 is true for sl2 remark 6 1 10 department of mathematics universiteitsplein 1 universitaire instelling antwer pen 2160 wilrijk belgium this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 44 27 utc all use subject to http about jstor org terms

a dynamic convergence analysis of blind equalization algorithms blind equalizers do not require a training sequence to start up or to restart after a communications breakdown making them particularly useful in applications such as broadcast and point to multipoint networks we study in parallel the dynamic convergence behavior of three blind equalization algorithms the multimodulus algorithm mma the constant modulus algorithm cma and the reduced constellation algorithm rca using a conditional gaussian approximation we first derive the theoretical mean squared error mse trajectory for mma this analysis leads to accurate but somewhat cumbersome expressions alternatively we apply a taylor series approximation to derive mse trajectories for mma cma and rca this approach yields simpler but somewhat less accurate expressions for the steady state operation however we derive even simpler formulas that accurately predict the asymptotic mse values we finally study the convergence rates of the three blind algorithms using their theoretical mse trajectories computer simulations and a laboratory experiment

ill convergence of godard blind equalizers in data communication systems the existence of stable undesirable equilibria for the godard algorithm is demonstrated through a simple autoregressive ar channel model these undesirable equilibria correspond to local but nonglobal minima of the underlying mean cost function and thus permit the ill convergence of the godard algorithms which are stochastic gradient descent in nature simulation results confirm predicted misbehavior for channel input of constant modulus it is shown that attaining the global minimum of the mean cost necessarily implies correct equalization a criterion is also presented for allowing a decision at the equalizer as to whether a global or nonglobal minimum has been reached

permanent polya problem for additive surjective maps abstract let m n f denote the set of square matrices of size n over a field f with characteristics different from two we say that the map f m n f m n f is additive if f a b f a f b for all a b m n f the main goal of this paper is to prove that for n 2 there are no additive surjective maps t m n f m n f such that per t a det a for all a m n f also we show that an arbitrary additive surjective map t m n f m n f which preserves permanent is linear and thus can be completely characterized

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

the failure of the artin rees property for the jacobson radical in prime noetherian rings abstract we provide an example of a prime noetherian ring whose jacobson radical fails to satisfy the artin rees ar property on either side the ring constructed is a finite module over its noetherian centre this settles a longstanding question

weak convergence and local stability properties of fixed step size recursive algorithms a recursive equation that subsumes several common adaptive filtering algorithms is analyzed for general stochastic inputs and disturbances by relating the motion of the parameter estimate errors to the behavior of an unforced deterministic ordinary differential equation ode the odes describing the motion of several common adaptive filters are examined in some simple settings including the least mean square lms algorithm and all three of its signed variants the signed regressor the signed error and the sign sign algorithms stability and instability results are presented in terms of the eigenvalues of a correlation like matrix this generalizes known results for lms signed regressor lms and signed error lms and gives new stability criteria for the sign sign algorithm the ability of the algorithms to track moving parameterizations can be analyzed in a similar manner by relating the time varying system to a forced ode the asymptotic distribution about the forced ode is an ornstein uhlenbeck process the properties of which can be described in a straightforward manner

new criteria for blind deconvolution of nonminimum phase systems channels a necessary and sufficient condition for blind deconvolution without observing the input of nonminimum phase linear time invariant systems channels is derived based on this condition several optimization criteria are proposed and their solution is shown to correspond to the desired response these criteria involve the computation only of second and fourth order moments implying a simple tap update procedure the proposed methods are universal in the sense that they do not impose any restrictions on the probability distribution of the unobserved input sequence it is shown that in several important cases e g when the additive noise is gaussian the proposed criteria are essentially unaffected

on invertible bimodules and automorphisms of noncommutative rings in this article we attempt to generalize the result that for a commu tative ring r the outer automorphism group of i automorphisms of m r is abelian of exponent n it is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings we also show that the automorphism group of an affine commutative domain of positive di mension acts faithfully on the spectrum of the domain we investigate other questions involving bimodules and automorphisms and extend a result of smith on the first weyl algebra as a fixed ring let r be a ring with 1 in this article we are interested in comparing the automorphism group of mn r the ring of x matrices over r and the automorphism group of r in particular if r is commutative it is a classical result that the outer automorphism group of r automorphisms of mn r is an abelian group of exponent dividing this can be restated in terms of invertible bimodules the result is equivalent to the easy fact that any projective rank one module x with nx free over a commutative ring has order dividing in the picent group of r see rz and ko in the noncommutative case there is still a well known relationship between automorphisms and the picard

picard groups and infinite matrix rings we describe a connection between the picard group of a ring with local units t and the picard group of the unital overring end tt using this connection we show that the three groups pic r pic fm r and pic rfm r are isomorphic for any unital ring r furthermore eaclh ele ment of pic rfm r arises from an automorphism of rfm r which yields an isomorphsm between pic rfm r and out rfm r as one applica tion we extend a classical result of rosenberg and zelinsky by showing that the group outr rfm r is abelian for any commutative unital ring r

the discriminant controls automorphism groups of noncommutative algebras we use the discriminant to determine the automorphism groups of some noncommutative algebras and we prove that a family of noncommutative algebras has tractable automorphism groups there is a long history and an extensive study of the automorphism groups of algebras determining the full automorphism group of an algebra is generally a no toriously difficult problem for example the automorphism group of t polynomial ring of three variables is not yet understood and a remarkable result in this direction is given by shestakov umirbaev su which shows the nagata automorphism is a wild automorphism since 1990s many researchers have been successfully comput ing the automorphism groups of interesting infinite dimensional noncommutative algebras including certain quantum groups generalized quantum weyl algebras skew polynomial rings and many more see alc ald and bj gtk sav which is only a partial list recently by using a rigidity theorem for quantum tori yakimov has proved the andruskiewitsch dumas conjecture and the launois lenagan conjecture in y1 y2 each of which determines the automorphism group of a family of quantized algebras with parameter q being not a root of unity a uniform approach to both the andruskiewitsch dumas conjecture and the launois lenagan conjecture is provided in a preprint by goodearl yakimov gy these beautiful results as well as others motivated us to look into the automorphism groups of noncommutative algebras

groups acting on affine algebras general actions of groups on commutative affine domains are studied we prove a finiteness theorem for orbits of ideals and an ergodic theorem inspired by results from the theories of group algebras and universal enveloping algebras this paper began as a reconsideration of ergodic theorems in the theory of group algebras 2 3 41 the philosophy of these results is the following if a is a lattice i e a group isomorphic to z7 and g is a group of automorphisms of a then g acts on the maximal spectrum of the group algebra k a thus g mixes up the points on a torus the ergodic theorems study the connections between the original action on a and the nature of orbits on the torus although one can obtain many analogues to classical theorems in ergodic theory in this way the program is ultimately insufficient one ought not confine the study to orbits of maximal ideals after all there are likely to be g invariant closed subsets of the torus which are not finite unions of closures of point orbits to account for these more complicated situations we introduce a simple minded but new notion to the theory of groups acting on rings it will turn out that this concept sheds light on old theorems concerning enveloping algebras of lie algebras and suggests new problems in commutative algebra for the remainder of the paper k denotes a commutative field and r is a commutative affine k algebra assume g is a group of k automorphisms of r we are particularly interested in the case that g is infinite the role played by maximal ideals above is taken over by g variant prime ideals a prime ideal p of r is g variant provided that p is maximal among all ideals q such that

ideal arithmetic in noetherian pi rings a domain satisfying the above equivalent characterisations is called a dedekind domain in the setting of non commutative orders the corresponding theory was axiomatised by asano 2 31 and further relined by michler t l and robson 1131 the starting point of this theory is condition b and indeed in modern terms the rings considered by asano can be thought of as prime noetherian rings satisfying a polynomial identity and condition fb however while condition a is often taken to be the definition of a commutative dedekind domain 10 p 294 15 p 2701 its implications in non commutative rings have not so far been studied it is the aim of this paper to do so one obstruction to following one of the standard proofs of the commutative theory is that if p p 1 p is an invertible product of prime ideals then it is not immediately clear that each pi is individually 475 0021 8693 89 3 00