on a system of klein gordon type equations with acoustic boundary conditions we prove the existence uniqueness and uniform stabilization of global solutions for a generalized system of klein gordon type equations with acoustic boundary conditions on a portion of the boundary and the dirichlet boundary condition on the rest

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

asymptotic stability of the wave equation on compact surfaces and locally distributed damping a sharp result this paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping described by u tt m u x g u t 0 on m 0 where m c r 3 is a smooth oriented embedded compact surface without boundary denoting by g the riemannian metric induced on m by r 3 we prove that for each 0 there exist an open subset v c m and a smooth function f m r such that meas v meas m e hessf g on v and inf f x 0 x v in addition we prove that if x a o 0 on an open subset m c m which contains m v and if g is a monotonic increasing function such that k s g s k s for all s 1 then uniform and optimal decay rates of the energy hold

nonlinear damped partial differential equations and their uniform discretizations abstract we establish sharp energy decay rates for a large class of nonlinearly first order damped systems and we design discretization schemes that inherit of the same energy decay rates uniformly with respect to the space and or time discretization parameters by adding appropriate numerical viscosity terms our main arguments use the optimal weight convexity method and uniform observability inequalities with respect to the discretization parameters we establish our results first in the continuous setting then for space semi discrete models and then for time semi discrete models the full discretization is then inferred from the previous results by adapting the ideas to deal with linear systems our results cover for instance the schrodinger equation with nonlinear damping the nonlinear wave equation the nonlinear plate equation the nonlinear transport equation as well as certain classes of equations with nonlocal terms

a unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi discretized vibrating damped systems the liapunov method is celebrated for its strength to establish strong decay of solutions of damped equations extensions to infinite dimensional settings have been studied by several authors see e g haraux 1991 11 and komornik and zuazua 1990 17 and references therein results on optimal energy decay rates under general conditions of the feedback is far from being complete the purpose of this paper is to show that general dissipative vibrating systems have structural properties due to dissipation we present a general approach based on convexity arguments to establish sharp optimal or quasi optimal upper energy decay rates for these systems and on comparison principles based on the dissipation property and interpolation inequalities in the infinite dimensional case for lower bounds of the energy we stress the fact that this method works for finite as well as infinite dimensional vibrating systems and as well as for applications to semi discretized nonlinear damped vibrating pde s a part of this approach has been introduced in alabau boussouira 2004 2005 1 2 in the present paper we identify a new simple and explicit criteria to select a class of nonlinear feedbacks for which we prove a simplified explicit energy decay formula comparatively to the more general but also more complex formula we give in alabau boussouira 2004 2005 1 2 moreover we prove optimality of the decay rates for this class in the finite dimensional case this class includes a wide range of feedbacks ranging from very weak nonlinear dissipation exponentially decaying in a neighborhood of zero to polynomial or polynomial logarithmic decaying feedbacks at the origin in the infinite dimensional case we establish a comparison principle on the energy of sufficiently smooth solutions through the dissipation relation this principle relies on suitable interpolation inequalities it allows us to give lower bounds for the energy of smooth initial data for the one dimensional wave equation with a distributed polynomial damping which improves haraux 1995 12 lower estimate of the energy for this case we also establish lower bounds in the multi dimensional case for sufficiently smooth solutions when such solutions exist we further mention applications of these various results to several classes of pde s namely the locally and boundary damped multi dimensional wave equation the locally damped plate equation and the globally damped coupled timoshenko beams system but it applies to several other examples furthermore we show that these optimal energy decay results apply to finite dimensional systems obtained from spatial discretization of infinite dimensional damped systems we illustrate these results on the one dimensional locally damped wave and plate equations discretized by finite differences and give the optimal energy decay rates for these two examples these optimal rates are not uniform with respect to the discretization parameter we also discuss and explain why optimality results have to be stated differently for feedbacks close to linear behavior at the origin

differential operators on an affine curve let x denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero denote by 3 x the sheaf of differential operators on x and 2 x f x 3 x the ring of global differential operators on x the following is established theorem 3 x is a finitely generated k algebra and a noetherian ring furthermore 2 x has a unique minimal non zero ideal j and 3 x j is a finite dimensional k algebra let x denoted the normalisation of x and n x x the projection map the main technique is to compare 2 x with 3 x

dispersive estimates for principally normal pseudodifferential operators the aim of these notes is to describe some recent re sults concerning dispersive estimates for principally normal pseu dodifferential operators the main motivation for this comes from unique continuation problems such estimates can be used to prove l q carleman inequalities which in turn yield unique continuation results for various partial differential operators with rough poten tials dispersive estimates are l q estimates for nonelliptic partial differ ential operators which are a consequence of the decay properties of their fundamental solutions these decay properties follow from spa tial spreading of the singularities of the solutions since solutions prop agate in directions conormal to the characteristic set of the operator this spreading can be related to nonzero curvatures of the characteristic set dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis various types of dispersive estimates are known to be true for op erators such as the wave operator the schrodinger operator and the linear kdv see ginibre velo 4 keel tao 11 they have proved to be useful in the study of nonlinear problems as well as of problems with unbounded potentials more recently similar estimates have been obtained for wave op erators with variable coefficients beginning with the smooth case in kapitanskii 10 mockenhaupt seeger and sogge 14 up to operators with c 2 coefficients in smith 15 and tataru 21 23 similar results were obtained for the schrodinger equation in staffilani tataru 19 c 2 coefficients and in burq gerard tzvetkov 1 smooth coeffic ients in the variable coefficient elliptic case one should also mention sogge s l q

mad subalgebras of rings of differential operators on curves we study the maximal abelian ad nilpotent mad subalgebras of the domains morita equivalent to the first weyl algebra we give a complete description both of the individual mad subalgebras and of the space of all such a surprising consequence is that this last space is independent of our results generalize some classic theorems of dixmier about the weyl algebra

a perturbation expansion for diffusion in a random velocity field we present a formal perturbation expansion for the large scale effective drift velocity and diffusion matrix of a medium with stationary random velocity v x and constant nonrandom diffusion matrix a on a small scale if mu denotes the mean of v x and we express v x as mu varepsilon u x then the expansion is in powers of varepsilon with mu u and a fixed we obtain explicit expressions up to second order in varepsilon which generalize the standard formulae to the case a ne 0

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

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

rings graded equivalent to the weyl algebra abstract we consider the first weyl algebra a in the euler gradation and completely classify graded rings b that are graded equivalent to a that is the categories gr a and gr b are equivalent this includes some surprising examples in particular we show that a is graded equivalent to an idealizer in a localization of a we obtain this classification as an application of a general morita type characterization of equivalences of graded module categories given a z graded ring r an autoequivalence f of gr r and a finitely generated graded projective right r module p we show how to construct a twisted endomorphism ring end r f p and prove theorem the z graded rings r and s are graded equivalent if and only if there are an autoequivalence f of gr r and a finitely generated graded projective right r module p such that the modules f n p generate gr r and s end r f p

spectral estimators that extend the maximum entropy and maximum likelihood methods the theory of best linear approximation in weighted l 2 spaces is used to obtain a general procedure the pdft for linearly reconstructing the fourier transform from sampled data the pdft can be used either directly to reduce sidelobe structure and to extrapolate the data or indirectly to obtain high resolution spectral estimators the direct and indirect pdft include as special cases many of the commonly used spectral techniques including burg s maximum entropy method capon s maximum likelihood method the spectral estimators based on bandlimited extrapolation the eigenvalue eigenvector methods for detecting sinusoids in noise pisarenko method schmidt s music eigenvector power beamforming and the best linear unbiased estimator blue for regression coefficients by exploiting their relationship to the linear pdft these nonlinear techniques can be analyzed in terms of linear approximation theory in addition to providing a unifying formulation for many different spectral estimators the pdft

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

asymptotic stability for a strongly coupled klein gordon system in an inhomogeneous medium with locally distributed damping abstract we consider a strongly coupled klein gordon system posed in an inhomogeneous medium with smooth boundary subject to a local damping distributed around a neighborhood of the boundary according to the geometric control condition we show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase space for this purpose refined microlocal analysis arguments are considered by exploiting ideas due to burq and gerard 16 by using sharp l p l q carleman estimates we prove a unique continuation property for coupled systems we also consider a strongly coupled system of wave equations with localized nonlinear damping acting only on one equation considering a compact riemannian manifold we show that the energy of the coupled system goes uniformly to zero

global nonexistence for abstract evolution equations with positive initial energy abstract in this paper we consider the problem of noncontinuation of solutions of the initial value problem for abstract evolution equations of the form pu tt q t u t a t u f t u t j 0 wherepandqare linear self adjoint operators anda t u andf t u are respectively a linear operator inu typically of differential type and a nonlinear driving force our principal concern is the noncontinuation or blow up of solutions when the initial energy is positive but appropriately bounded

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

a parametric family of subalgebras of the weyl algebra i structure and automorphisms an ore extension over a polynomial algebra f x is either a quantum plane a quantum weyl algebra or an infinite dimensional unital associative algebra ah generated by elements x y which satisfy yx xy h where h f x we investigate the family of algebras ah as h ranges over all the polynomials in f x when h 6 0 these algebras are subalgebras of the weyl algebra a1 and can be viewed as differential operators with polynomial coefficients we give an exact description of the automorphisms of ah over arbitrary fields f and describe the invariants in ah under the automorphisms we determine the center normal elements and height one prime ideals of ah localizations and ore sets for ah and the lie ideal ah ah we also show that ah cannot be realized as a generalized weyl algebra over f x except when h f in two sequels to this work we completely describe the derivations and irreducible modules of ah over any field

stabilization and control for the nonlinear schrodinger equation on a compact surface in this paper we study the stabilization property and the exact controllability for the nonlinear schrodinger equation on a two dimensional compact riemannian manifold without boundary we use a pseudo differential dissipation the proofs are based on a result of propagation of singularities and on recent dispersion estimates strichartz type inequalities due to n burq p gerard and n tzvetkov

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

uniform decay rate estimates for schrodinger and plate equations with nonlinear locally distributed damping on a compact n dimensional riemannian manifold m g we establish uniform decay rate estimates for the linear schrodinger and plate equations subject to an internal nonlinear damping locally distributed on the manifold our approach can be also employed for other equations provided that inverse inequality for the linear model occurs in the particular case of the wave equation where the well known geometric control condition gcc is equivalent to the observability inequality our method generalizes the results due to cavalcanti et al 2010 2009 9 10 regarding the optimal choice of dissipative regions

fixed rings of generalized weyl algebras abstract we study actions by filtered automorphisms on classical generalized weyl algebras gwas in the case of a defining polynomial of degree two we prove that the fixed ring under the action of a finite cyclic group of filtered automorphisms is again a classical gwa extending a result of jordan and wells partial results are provided for the case of higher degree polynomials in addition we establish a version of auslander s theorem for finite cyclic groups of filtered automorphisms acting on classical gwas

asymptotic stability of the wave equation on compact manifolds and locally distributed damping a sharp result let m g be a n dimensional n geqq 2 compact riemannian manifold with boundary where g denotes a riemannian metric of class c this paper is concerned with the study of the wave equation on m g with locally distributed damping described by left begin array l u tt delta bf g u a x g u t 0 quad hbox on thinspace m times left 0 infty right u 0 hbox on partial m times left 0 infty right end array right where m represents the boundary of m and a x g ut is the damping term the main goal of the present manuscript is to generalize our previous result in cavalcanti et al trans ams 361 9 4561 4580 2009 treating the conjecture in a more general setting and extending the result for n dimensional compact riemannian manifolds m g with boundary in two ways i by reducing arbitrarily the region m ast subset m where the dissipative effect lies this gives us a totally sharp result with respect to the boundary measure and interior measure where the damping is effective ii by controlling the existence of subsets on the manifold that can be left without any dissipative mechanism namely a precise part of radially symmetric subsets an analogous result holds for compact riemannian manifolds without boundary

dixmier groups and borel subgroups abstract in this paper we study a family g n n 0 of infinite dimensional ind algebraic groups associated with algebras morita equivalent to the weyl algebra a 1 c we give a geometric presentation of these groups in terms of amalgamated products generalizing classical theorems of dixmier and makar limanov our main result is a classification of borel subgroups of g n for all n we show that the conjugacy classes of non abelian borel subgroups of g n are in bijection with the partitions of n furthermore we prove an infinite dimensional analogue of the classical theorem of steinberg 52 that characterizes borel subgroups in purely group theoretic terms combined together the last two results imply that the g n are pairwise non isomorphic as abstract groups this settles an old question of stafford 51

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

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

asymptotics of solute dispersion in periodic porous media the concentration c bf x t of a solute in a saturated porous medium is governed by a second order parabolic equation partial c partial t u 0 bf b cdot nabla c tfrac 1 2 sum d ij partial 2 c partial x i partial x j in the case that bf b is periodic and divergence free and d ij are constants and d ij positive definite the concentration is asymptotically gaussian for large times this article analyzes the dependence of the dispersion matrix bf k of the limiting gaussian distribution on the velocity parameter u 0 and the period a it is shown that each coefficient k ii is asymptotically quadratic in au 0 if b i bar bar b j has a nonzero component in the null space of bf b cdot nabla and asymptotically constant in au 0 if b i bar bar b j belongs to the range of bf b cdot nabla it is shown in a more general context that bf k depends only on au 0 an asymptotic expansion of the cramer edgeworth typ

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

uniform decay rates for the wave equation with nonlinear damping locally distributed in unbounded domains with finite measure this paper is concerned with the study of the uniform decay rates of the energy associated with the wave equation with nonlinear damping locally distributed u tt delta u a x g u t 0 hbox in omega times 0 infty subject to dirichlet boundary conditions where omega subset mathbb r n n geq 2 is an unbounded open set with finite measure and unbounded smooth boundary partial omega gamma the function a x responsible for the localized effect of the dissipative mechanism is assumed to be nonnegative and essentially bounded and in addition a x geq a 0 0 hbox a e in omega where omega omega cup x in omega x r r 0 and omega is a neighborhood in omega of the closure of partial omega cap b r where b r x in omega x r

rings of operators on modules over commutative rings and their right ideals suppose that there is an inclusion ofk algebrasr e endkmwithrcommutative andenon commutative we introduce and impose conditions under which the finitely generated essential right ideals ofemay be classified in terms ofk submodules ofm this yields a classification of the domains morita equivalent toewheneis a noetherian domain for example a special case of our results is theorem let r be a commutative noetherian k algebra which is domain let e be a simple ore extension of r of the form r x x 1 or r x in the latter case we must also assume r 3 then for a certain sublatticelof the lattice of k submodules of r a every non zero right ideal of e is isomorphic to one of the form formula for some v l b every domain morita equivalent to e is isomorphic to formula for some v l conversely if r is dedekind then e v is morita equivalent to e for v l