on the ideals of a noetherian ring we construct various examples of noetherian rings with peculiar ideal structure for example there exists a noetherian domain r with a minimal nonzero ideal 1 such that r i is a commutative polynomial ring in n variables and a noetherian domain s with a second layer clique that is not locally finite the key step in the construction of these rings is to idealize at a right ideal i in a noetherian domain t such that t l is not artinian introduction the aim of this paper is to construct several examples concerned with the ideal structure of a noetherian ring we will delay discussion of their relevance until we are ready to construct them in 2 3 and 4 and simply describe their properties at this stage however for the most part these examples complement the positive results in the author s survey article 12 example 1 see theorem 2 3 let m be a positive integer there exists a noetherian domain r with a unique minimal nonzero ideal i such that r i is a polynomial ring in m variables in particular r has infinitely many prime ideals example 2 theorem 3 2 there exists a prime noetherian ring r with goldie rank two written grk r 2 yet for each n c n there exists a prime ideal qn of r with grk r qn 2n 1 further nqn 0 example 3 theorem 4 1 there exists a noetherian ring r and two prime ideals p1 and p2 of r such that there exists an ideal link between p1 and p2 but such that pi and p2 do not belong to the same second layer clique example 4 theorem 4 4 there exists a noetherian domain r and a second layer clique of prime ideals of r that is not locally finite this clique also satisfies the second layer condition the connecting theme between these examples and the crucial step in their construction is the concept of the idealizer i i f e s fi c i of a right ideal i in a noetherian ring s of course since robson s seminal paper 8 this has been a well known and productive method of producing interesting examples however in previous examples s i has been artinian unfortunately this only produces finitely many new ideals whereas the present examples are concerned with the properties of an infinite set of prime ideals thus we are forced to idealize at a right ideal i such that s i is not artinian in general such an idealizer will not be left or right received by the editors june 11 1984 and in revised form july 30 1984 1980 mathematics subject classification primary 16a33 16a66 secondary 16a08

persistence in nonautonomous predator prey systems with infinite delays this paper studies the general nonautonomous predator prey lotka volterra systems with infinite delays the sufficient and necessary conditions of integrable form on the permanence and persistence of species are established a very interesting and important property of two species predator prey systems is discovered that is the permanence of species and the existence of a persistent solution are each other equivalent particularly for the periodic system with delays applying these results the sufficient and necessary conditions on the permanence and the existence of positive periodic solutions are obtained some well known results on the nondelayed periodic predator prey lotka volterra systems are strongly improved and extended to the delayed case

calderon zygmund estimates for generalized double phase problems abstract we prove calderon zygmund type estimates for distributional solutions to non uniformly elliptic equations of generalized double phase type in divergence form in particular we provide sharp conditions on the nonlinear operators to establish the calderon zygmund type estimates

calderon zygmund estimates and non uniformly elliptic operators abstract we consider a class of non uniformly nonlinear elliptic equations whose model is given by div d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where p q and a x 0 and establish the related nonlinear calderon zygmund theory in particular we provide sharp conditions under which the natural and optimal calderon zygmund type result f p a x f q l loc d u p a x d u q l loc holds for every 1 these problems naturally emerge as euler lagrange equations of some variational integrals introduced and studied by marcellini 41 and zhikov 53 in the framework of homogenisation and lavrentiev phenomenon

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

global attractivity of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion by employing the continuation theorem of coincidence degree theory the existence of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion is established further by constructing a lyapunov functional and using the result of the existence of positive periodic solution the attractivity of a positive periodic solution for above system is obtained

on the operator l f f log f abstract the present note deals with the operator l h 1 r n d r n which takes a given element f of the hardy space to the function f log f in general this function need not be locally integrable nevertheless due to peculiar cancellations of large positive and negative terms in the integral f log f with c 0 r n we are able to give meaning to f log f as a schwartz distribution we find several alternatives for this interpretation of f log f

harmless delays for uniform persistence abstract this paper studies a two dimensional predator prey system with a finite number of discrete delays our purpose is to demonstrate that the time delays are harmless for uniform persistence of the solutions of the system the results are obtained by constructing suitable persistence functionals

calderon zygmund estimates for elliptic double phase problems with variable exponents abstract we study a nonlinear elliptic double phase problem with variable exponents to prove calderon zygmund estimates under minimal regularity requirements on the nonlinearities

a class of noncommutative projective surfaces let a k a 1 a 2 be a connected graded noetherian k algebra that is generated in degree one over an algebraically closed field k suppose that the graded quotient ring q a has the form q a k y t t 1 sigma where sigma is an automorphism of the integral projective surface y then we prove that a can be written as a naive blowup algebra of a projective surface x birational to y this enables one to obtain a deep understanding of the structure of these algebras for example generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme this is despite the fact that the simple objects in the quotient category qgr a will always be in 1 1 correspondence with the closed points of the scheme x

global gradient estimates for the borderline case of double phase problems with bmo coefficients in nonsmooth domains abstract we consider a double phase problem with bmo coefficient in divergence form on a bounded nonsmooth domain the problem under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the position which describes a feature of strongly anisotropic materials we obtain the global calderon zygmund type estimates for the distributional solution in the case that the associated nonlinearity has a small bmo and the boundary of the domain is sufficiently flat in the reifenberg sense

field of fractions of certain quantum algebras abstract in this paper we show that the field of fractions of certain algebras appearing in the theory of quantum groups are isomorphic to the field of fractions of quantum n space when the parameter q is not a root of 1 more precisely the quantum serre relations of type a n and the relations defining the quantum weyl algebra induce q communtation relations in the field of fractions also we study the problem of separation of these skew fields using the standard embeddings in non commutative series

elliptic equations with bmo nonlinearity in reifenberg domains given p 2 8 we obtain the global w1 p estimate for the weak solution of a boundary value problem for an elliptic equation with bmo nonlinearity in a reifenberg domain assuming that the nonlinearity has sufficiently small bmo seminorm and that the boundary of the domain is sufficiently flat

the calderon zygmund property for quasilinear divergence form equations over reifenberg flat domains abstract the results by palagachev 2009 3 regarding global holder continuity for the weak solutions to quasilinear divergence form elliptic equations are generalized to the case of nonlinear terms with optimal growths with respect to the unknown function and its gradient moreover the principal coefficients are discontinuous with discontinuity measured in terms of small b m o norms and the underlying domain is supposed to have fractal boundary satisfying a condition of reifenberg flatness the results are extended to the case of parabolic operators as well

persistence and global stability for nonautonomous predator prey system with diffusion and time delay abstract in this paper a nonautonomous predator prey model with diffusion and continuous time delay is studied where all parameters are time dependent the system which is composed of two lotka volterra patches has two species one can diffuse between two patches but the other is confined to one patch and cannot diffuse it is proved that the system is uniformly persistent under appropriate conditions furthermore sufficient conditions are established for global stability of the system

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

gradient estimates for double phase problems with irregular obstacles abstract an irregular obstacle problem with non uniformly elliptic operator in divergence form of p q growth is studied we find an optimal regularity for such a double phase obstacle problem by essentially proving that the gradient of a solution is as integrable as both the gradient of the assigned obstacle function and the associated nonhomogeneous term in the divergence calderon zygmund type estimates are also obtained under minimal regularity requirements of the prescribed data

harmless delays and global attractivity for nonautonomous predator prey system with dispersion abstract in this paper we consider a nonautonomous predator prey model with dispersion and a finite number of discrete delays the system consists of two lotka volterra patches and has two species one can disperse between two patches but the other is confined to one patch and cannot disperse our purpose is to demonstrate that the time delays are harmless for uniform persistence of the solutions of the system furthermore we establish conditions under which the system admits a positive periodic solution which attracts all solutions

the simple modules of certain generalized crossed products abstract we consider a dedekind domaindand az graded ring r i z r i withr0 dand eachri dvibeing a freed module of rank 1 the structure ofris described by an automorphism ofdand a generalized 2 cocyclec z z dnot necessarily taking its values in the units ofd the aim of this paper is to classify the simpler modules say in the case wherec i j 0 for alli j z we also deal with this problem in then graded case as a consequence we obtain a description of the simple modules of some classical algebras and of generalized weyl algebras

harnack inequalities for double phase functionals we prove a harnack inequality for minimisers of a class of non autonomous functionals with non standard growth conditions they are characterised by the fact that their energy density switches between two types of different degenerate phases

scalar minimizers with fractal singular sets lack of regularity of local minimizers for convex functionals with non standard growth conditions is considered it is shown that for every 0 there exists a function ac such that the functional admits a local minimizer uw1 p whose set of non lebesgue points is a closed set with dim n p and where 1 p n n q

regularity for double phase problems under additional integrability assumptions abstract we study the regularity of quasi minimizers or minimizers of functionals of double phase type in recent years baroni colombo and mingione have investigated regularity theory for bounded minimizers in baroni et al 2015 2018 colombo and mingione 2015 in this paper we assume the minimizers only belong to l space

regularity for minimizers of double phase functionals with mild transition and regular coefficients abstract we prove sharp regularity results for minimizers of the functional p w b x w d w p a x d w p log e d w d x with w w 1 1 p 1 a l a 0 and 0 b l p is a double phase functional with mild transition between d u p and d u p log e d u first under suitable conditions on the moduli of continuity of a and b we prove that local minimizers are of class c 0 for every 0 1 then that they are of class c 1 for some 0 provided the functions a and b are holder continuous

weighted and regularity estimates for nonlinear equations on reifenberg flat domains abstract global weighted l p estimates are obtained for the gradient of solutions to nonlinear elliptic dirichlet boundary value problems over a bounded nonsmooth domain morrey and holder regularity of solutions are also established as a consequence these results generalize various existing estimates for nonlinear equations the nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation the boundary of the domain on the other hand may exhibit roughness but assumed to be sufficiently flat in the sense of reifenberg our approach uses maximal function estimates and vitali covering lemma and also known regularity results of solutions to nonlinear homogeneous equations

uniform persistence in functional differential equations abstract in this paper we investigate the question of uniform persistence for retarded functional differential equations by utilizing liapunov like functions razumikhin techniques and differential inequalities we are able to establish criteria for uniform persistence analogous to those obtained by others for ordinary differential equations difference equations and reaction diffusion equations we apply these criteria to some well known biological models with delay our results indicate that the conditions which guarantee the existence of an interior equilibrium are enough to ensure uniform persistence moreover these conditions are equivalent to uniform persistence for the cases without delay as well

some noncommutative minimal surfaces abstract in the ongoing programme to classify noncommutative projective surfaces connected graded noetherian domains of gelfand kirillov dimension three a natural question is to determine the minimal models within any birational class in this paper we show that the generic noncommutative projective plane corresponding to the three dimensional sklyanin algebra as well as noncommutative analogues of p 1 p 1 and the more general van den bergh quadrics satisfy very strong minimality conditions translated into an algebraic question where one is interested in a maximality condition we prove the following result theorem a let r be a sklyanin algebra or a van den bergh quadric that is infinite dimensional over its centre and let a r be any connected graded noetherian maximal order with the same graded quotient ring as r then up to taking veronese rings a is isomorphic to r let t be an elliptic algebra that is the coordinate ring of a noncommutative surface containing an elliptic curve then under an appropriate homological condition we prove that every connected graded noetherian overring of t is obtained by blowing down finitely many lines line modules of self intersection 1

an imbedding theorem for musielak sobolev spaces abstract let be a bounded domain in r n and a c 1 0 0 be a musielak orlicz function we established an imbedding theorem for the musielak sobolev space w 1 a of the form w 1 a l a where a is the sobolev conjugate function of a a musielak orlicz function defined in the paper this theorem extends the imbedding theorem established by donaldson and trudinger for orlicz sobolev spaces to the case of musielak sobolev spaces

global regularity in lorentz spaces for nonlinear elliptic equations with lp logl growth abstract we prove a global calderon zygmund type estimate in the framework of lorentz spaces for the gradients of weak solutions of nonlinear elliptic equations with l p log l growth more precisely by considering the zero dirichlet problem of the elliptic equation div a x d u div h x f x with a x d a x p x log e for any 0 a x 0 0 and h x f p x 1 log e f we obtain x f l t q x d u l t q for t 1 and q 0 where x p x log e under the assumptions that the function a has a small bounded mean oscillation seminorm p x satisfies strong log holder continuity and the underlying domain is reifenberg flat

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

holder regularity for nonlocal double phase equations abstract we prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro differential equations whose leading operator switches between two different types of fractional elliptic phases according to the zero set of a modulating coefficient a a the model case is driven by the following nonlocal double phase operator u x u y p 2 u x u y x y n s p d y a x y u x u y q 2 u x u y x y n t q d y where q p and a 0 our results do also apply for inhomogeneous equations for very general classes of measurable kernels by simply assuming the boundedness of the modulating coefficient we are able to prove that the solutions are holder continuous whereas similar sharp results for the classical local case do require a to be holder continuous to our knowledge this is the first regularity result for nonlocal double phase problems

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

global gradient estimates for a borderline case of double phase problems with measure data abstract we obtain a global calderon zygmund estimate for a borderline case of double phase problems with measure data in terms of the 1 fractional maximal function of the measure

irreducible representations of the 4 dimensional sklyanin algebra at points of infinite order abstract in 1982 sklyanin funct anal appl 16 1982 27 34 defined a family of graded algebras a e depending on an elliptic curve e and a point e which is not 4 torsion basic properties of these algebras were established in smith and stafford compositio math 83 1992 259 289 and a study of their representation theory was begun in levasseur and smith bull soc math france121 1993 35 90 the present paper classifies the finite dimensional simple a modules when is a point of infinite order sklyanin funct anal appl 17 1983 273 284 defines for each k n a representation of a in a certain k dimensional subspace of theta functions of order 2 k 1 we prove that these are irreducible representations and that any other simple module is obtained by twisting one of these by an automorphism of a the automorphism group of a is explicitly computed the method of proof relies on results in levasseur and smith in particular it is proved that every finite dimensional simple module is a quotient of a line module an important part of the analysis is a determination of the 1 critical a modules and the fact that such a module is equivalent to a quotient of a line module by a shifted line module

permanence and extinction for a nonlinear diffusive predator prey system abstract in this paper we consider a non autonomous predator prey system in a poor patchy environment under the assumption that the intrinsic growth rates of the species may be negative sufficient conditions are obtained for the permanence and extinction of the system considered

divergence form operators in reifenberg flat domains we study the boundary regularity of solutions of elliptic operators in divergence form with c 0 coefficients or operators which are small perturbations of the laplacian in non smooth domains we show that as in the case of the laplacian there exists a close relationship between the regularity of the corresponding elliptic measure and the geometry of the domain

noncommutative graded domains with quadratic growth letk be an algebraically closed field and letr be a finitely generated connected gradedk algebra which is a domain of gelfand kirillov dimension two write the graded quotient ringq r ofr asd z z 1 for some automorphism of the division ringd we prove thatd is a finitely generated field extension ofk of transcendence degree one moreover we describer in terms of geometric data ifr is generated in degree one then up to a finite dimensional vector space r is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curvey this implies in particular thatr is noetherian thatr is primitive when and thatr is a finite module over its centre when ifr is not generated in degree one thenr will still be noetherian and primitive if has infinite order butr need not be noetherian when has finite order

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 convexity of musielak orlicz sobolev spaces and applications abstract this paper deals with uniform convexity of musielak orlicz sobolev spaces and its applications to variational problems some sufficient conditions and examples for uniform convexity of musielak orlicz sobolev spaces are given some special properties relative to the uniformly convex modular for uniformly convex musielak orlicz sobolev spaces are presented as an application of these abstract results the local minimizers and the mountain pass type critical point of an integral functional with more complicated growth than the p x growth are studied

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f

riesz potential estimates for a class of double phase problems abstract a double phase problem with a bounded borel measure on the right hand side is studied we prove an optimal pointwise gradient estimate for such a measure data problem via riesz potentials of the measure under log dini continuity assumption on the modulating coefficients as a consequence we find an optimized c 1 regularity criterion

regularity for multi phase variational problems abstract we prove c 1 regularity for local minimizers of the multi phase energy w d w p a x d w q b x d w s d x under sharp assumptions relating the couples p q and p s to the holder exponents of the modulating coefficients a and b respectively

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

derivations of a parametric family of subalgebras of the weyl algebra abstract 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 a h generated by elements x y which satisfy y x x y h where h f x when h 0 the algebra a h is subalgebra of the weyl algebra a 1 and can be viewed as differential operators with polynomial coefficients this paper determines the derivations of a h and the lie structure of the first hochschild cohomology group hh 1 a h der f a h inder f a h of outer derivations over an arbitrary field in characteristic 0 we show that hh 1 a h has a unique maximal nilpotent ideal modulo which hh 1 a h is 0 or a direct sum of simple lie algebras that are field extensions of the one variable witt algebra in positive characteristic we obtain decomposition theorems for der f a h and hh 1 a h and describe the structure of hh 1 a h as a module over the center of a h

hochschild and cyclic homology of yang mills algebras the aim of this article is to compute the hochschild and cyclic homology groups of the yang mills algebras ym n n n 2 defined by a connes and m dubois violette in cd1 contin uing thus the study of these algebras that we have initiated in hs the computation involves the use of a spectral sequence associated to the natural filtration on the universal enveloping algebra ym n provided by a lie idealtym n inym n which is free as lie algebra 2000 mathematics subject classification 16e40 16s32 17b56 70s15 81t13