regularity of invariant measures on finite and infinite dimensional spaces and applications abstract in this paper we prove new results on the regularity i e smoothness of measures solving the equation l 0 for operators of type l b on finite and infinite dimensional state spaces e in particular we settle a conjecture of i shigekawa in the situation where h is the gross laplacian e h is an abstract wiener space and b id e v where v takes values in the cameron martin space h using gross logarithmic sobolev inequality in an essential way we show that is always absolutely continuous w r t the gaussian measure and that the square root of the density is in the malliavin test function space of order 1 in l 2 furthermore we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for l 0 these include results on the inverse problem i e we give conditions ensuring that b is the vector logarithmic derivative of a measure we also prove necessary and sufficient conditions for to be symmetrizing i e l is symmetric on l 2 finally a substantial part of this work is devoted to the uniqueness of symmetrizing measures for l we characterize the cases where we have uniqueness by the irreducibility of the associated classical dirichlet forms

classical dirichlet forms on topological vector spaces the construction of the associated diffusion process given a minimal classical dirichlet form onl 2 e we construct the associated diffusion process heree is a locally convex topological vector space and is a not necessarily quasi invariant probability measure one the construction is carried out under certain assumptions one and which can be easily verified in many examples in particular we explicitly apply our results to time zero and space time quantum fields with or with out cut off

classical dirichlet forms on topological vector spaces closability and a cameron martin formula abstract we prove a sufficient condition for the closability of classical dirichlet forms on l 2 e which is also necessary if all components of the dirichlet form are closable here e is a locally convex topological vector space and a not necessarily quasi invariant probability measure on e the same condition is shown to imply the existence of a closed extension whose domain can be described explicitly in the special case where is quasi invariant with respect to certain vectors in e our result generalises previous theorems on closability in addition we prove a cameron martin type formula for a large class of measures if e is finite dimensional our characterisation of closability is the analogue of the corresponding one dimensional result applications to quantum fields and the connection with the well studied case of abstract wiener spaces are discussed

quasi invariance and reversibility in the fleming viot process fleming viot processes with mutation selection and recombination are studied their reversible distributions are shown to be characterized as quasi invariant measures with a cocycle given in terms of the mutation operator the selection intensity and the recombination kernel by using this we derive not only a necessary and sufficient condition for the fleming viot process to be reversible but also identify the reversible distributions in the reversible case

duo rings and ore extensions abstract we show that in an ore polynomial ring every left ideal is two sided only in the trivial commutative case it is unknown whether the analogous statement about right ideals is true we establish a variety of restrictions on any putative counterexample

localization and ideal theory in iterated differential operator rings abstract in this paper we study primality hypercentrality simplicity and localization and the second layer condition in skew enveloping algebras and iterated differential operator rings we give sufficient conditions for the skew enveloping algebra of a nilpotent lie algebra with coefficient ring containing the rational numbers to be a simple ring and we give necessary and sufficient conditions in the case that the lie algebra is abelian our main results show that if l is a finite dimensional solvable lie algebra over a field k of characteristic zero and r is an artinian or a commutative noetherian algebra over k then the skew enveloping algebra r u l satisfies the second layer condition we discuss consequences of this for localization and use the localization theory to state a classical krull dimension versus global dimension inequality when k is uncountable

euclidean gibbs measures on loop lattices existence and a priori estimates we present a new method to prove existence and uniform a priori estimates for euclidean gibbs measures corresponding to quantum anharmonic crystals it is based first on the alternative characterization of gibbs measures in terms of their logarithmic derivatives through integration by parts formulas and second on the choice of appropriate lyapunov functionals

uniqueness and global markov property for euclidean fields the case of general polynomial interactions we give a general method for proving uniqueness and global markov property for euclidean quantum fields the method is based on uniform continuity of local specifications proved by using potential theoretical tools and exploitation of a suitable fkg order structure we apply this method to give a proof of uniqueness and global markov property for the gibbs states and to study extremality of gibbs states also in the case of non uniqueness in particular we prove extremality for 2 4 also in the case of non uniqueness and global markov property for weak coupling 2 4 which solves a long standing problem uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field

ray holder continuity for fractional sobolev spaces in infinite dimensions and applications we prove holder continuity on rays in the direction of vectors in the generalized cameron martin space for functions in sobolev spaces in lp of fractional order 1 over infinite dimensional linear spaces the underlying measures are required to satisfy some easy standard structural assumptions only apart from wiener measure they include gibbs measures on a lattice and euclidean interacting quantum fields in infinite volume a number of applications e g to the two dimensional polymer measure are presented in particular irreducibility of the dirichlet form associated with the latter measure is proved without restrictions on the coupling constant

identities ore extensions and character skew derivations following 17 a derivation is called quasi algebraic if there exist a ai q such that k x i 1k 1ai i x ax k x a 0 for x q let k be the minimal such integer and g0 g1 mutually outer automorphisms of q let f zij be a generalized polynomial in variables zij where i 0 and 0 j k it is shown in 6 that if f gi j x or f jgi x is an identity then so is f zij we generalize this to sets of skew derivations in theorems 1 and 2 in theorem 3 we apply these results to the ore extensions of several variables of 2 and in theorem 4 to the character hopf algebras of the character skew derivations of 12

localization and ideal theory in noetherian strongly group graded rings abstract we study primality hypercentrality simplicity and localization and the second layer condition in skew groups rings and group graded rings we give necessary and sufficient conditions for the skew group ring of a torsion free nilpotent group to be a simple ring and if the coefficient ring is commutative we give necessary and sufficient conditions for the skew group ring of an abelian group to be simple our method involves showing certain group graded rings are hypercentral our main results show that if g is a polycyclic by finite group and r is an artinian ring or a commutative noetherian ring then a strongly g graded ring with base ring r satisfies the second layer condition we discuss consequences of this for localization in such rings

on the structure of semi invariant polynomials in ore extensions let d be an x outer s derivation of a prime ring r where s is an automorphism of r the following is proved among other things the degree of the minimal semi invariant polynomial of the ore extension r x s d is if charr 0 and is pk for some k 0 if charr p 2 where is the least integer 1 such that s ds d is x inner a similar result holds for cv polynomials these are done by introducing the new notion of k basic polynomials for each integer k 0 which enable us to analyze semi invariant polynomials inductively

mosco convergence of dirichlet forms in infinite dimensions with changing reference measures let e be an infinite dimensional locally convex space let n be a weakly convergent sequence of probability measures on e and let en be a sequence of dirichlet forms on e such that en is defined on l2 n general sufficient conditions for mosco convergence of the gradient dirichlet forms are obtained applications to gibbs states on a lattice and to the gaussian case are given weak convergence of the associated processes is discussed

a hierarchical model of quantum anharmonic oscillators critical point convergence a hierarchical model of interacting quantum particles performing anharmonic oscillations is studied in the euclidean approach in which the local gibbs states are constructed as measures on infinite dimensional spaces the local states restricted to the subalgebra generated by fluctuations of displacements of particles are in the center of the study they are described by means of the corresponding temperature green matsubara functions the result of the paper is a theorem which describes the critical point convergence of such matsubara functions in the thermodynamic limit

ergodicity ofl2 semigroups and extremality of gibbs states we extend the classical results of holley stroock on the characterization of extreme gibbs states for the ising model in terms of the irreducibility resp ergodicity of the corresponding glauber dynamics to the case of lattice systems with unbounded linear spin spaces we first develop a general framework to discuss questions of this type using classical dirichlet forms on infinite dimensional state spaces and their associated diffusions we then describe concrete applications to lattice models with polynomial interactions i e the discretep d models of euclidean quantum field theory in addition we prove the equivalence of extremality and shift ergodicity for tempered gibbs states of these models and also discuss this question in the general framework

large deviations w r t quasi every starting point for symmetric right processes on general state spaces in the work of donsker and varadhan fukushima and takeda and that of deuschel and stroock it has been shown that the lower bound for the large deviations of the empirical distribution of an ergodic symmetric markov process is given in terms of its dirichlet form we give a short proof generalizing this principle to general state spaces that include in particular infinite dimensional and non0metrizable examples our result holds w r t quasi every starting point of the markov process moreover we show the corresponding weak upper bound w r t quasi every starting point

dirichlet operators on loop spaces essential self adjointness and log sobolev inequality for each 0 1 and potential function v rd r we consider the dirichlet form e and the associated dirichlet operator h for the gibbs measure on the loop space e c 0 1 rd 0 1 the gibbs measure is related to the gibbs state of the quantum anharmonic oscillator with the potential v via the feynman kac formula we formulate dirichlet forms in the framework of rigged hilbert spaces which are related to the loop space e we then give an approximate criterion for the essential self adjointness of dirichlet operators associated with dirichlet forms given by probability measures on hilbert spaces under appropriate conditions on the potential we apply the approximate criterion to show that the dirichlet operator h is essentially self adjoint on the domain of smooth cylinder functions in addition if the potential satisfies a uniform convexity condition we prove that the dirichlet operator h has a gap at the lower end of spectrum we also show that the gibbs measure satisfie

analysis and geometry on configuration spaces the gibbsian case using a natural riemannian geometry like structure on the configuration space over rd we prove that for a large class of potentials the corresponding canonical gibbs measures on 1 can be completely characterized by an integration by parts formula that is if is the gradient of the riemannian structure on one can define a corresponding divergence div such that the canonical gibbs measures are exactly those measures for which and div are dual operators onl2 one consequence is that for such the corresponding dirichlet forms e are defined in addition each of them is shown to be associated with a conservative diffusion process on with invariant measure the corresponding generators are extensions of the operator div the diffusions can be characterized in terms of a martingale problem and they can be considered as a brownian motion on perturbed by a singular drift another main result of this paper is the following if is a canonical gibbs measure then it is extreme or a pure phase if and only if the corresponding weak sobolev spacew1 2 on is irreducible as a consequence we prove that for extreme canonical gibbs measures the above mentioned diffusions are time ergodic in particular this holds for tempered grand canonical gibbs measures ruelle measures provided that the activity constant is small enough we also include a complete discussion of the free case i e 0 where the underlying space rdis even replaced by a riemannian manifoldx

the influence of ideal structure on representation theory the purpose of this paper is to explore the representation theoretic consequences of the two sided ideal structure of a noetherian ring with special emphasis on an additional combinatorial structure on the prime spectrum of the ring the graph of links we obtain existence theorems for extensions and for rings satisfying the strong second layer condition we obtain extensive information about the indecomposable injective modules particularly detailed results are obtained when the ring is polynormal

identities with a single skew derivation abstract let r be a prime ring with extended centroid c and a continuous skew derivation of r we define the notion of k polynomials which in the case that is an ordinary derivation reduces to polynomials of the form x p n 1 x p n 1 n x where i c it is shown that all generalized identities with are consequences of gpis of r and an identity in the form x x m b b x where x is a k polynomial of minimal possible order m

minimal quasi algebraic relations of skew derivations in characteristic p 2 abstract let r be a prime ring of char r p 2 the monic minimal quasi algebraic relation of an ordinary derivation d of r takes the form d p s i 0 s 1 i d p i ad a where i are in the extended centroid of r and where a is in the symmetric martindale quotient ring of r among other things we extend this result to an arbitrary skew derivation d this is explained in details in 0 below

uniqueness of generalized schrodinger operators and applications abstract we prove necessary and sufficient conditions on r d r 0 a e so that the operator s 2 1 w1 n has exactly one self adjoint extension on w1 p which generates a sub markovian semigroup on w1 n these conditions are shown to be always fulfilled if d 1 and are verified for a large class of functions if d 1 we also prove an infinite dimensional analogue of this result where r d is replaced by some topological vector space e and the lebesgue measure dx by a gaussian measure the role of is taken by some ornstein uhlenbeck operator with general linear drift and dom s bounded smooth cylinder functions both results are derived as special cases from a general perturbation result we also give the corresponding probabilistic version i e we prove uniqueness for the associated martingale problem we present in some detail applications to the case where is a ground state of a hamiltonian in quantum field theory settling a problem which has been open for some time furthermore we apply our results to prove uniqueness in distribution of the diffusion first constructed by g jona lasinio and p k mitter for the stochastic quantization of f 2 field theory

a priori estimates for symmetrizing measures and their applications to gibbs states abstract we prove existence and uniform a priori estimates for tempered gibbs states of certain classical lattice systems with unbounded spins nonharmonic pair potentials and infinite radius of interaction we use an alternative characterization of gibbs measures in terms of their radon nikodym derivatives w r t local shifts of the configuration space and corresponding integration by parts formulas

on the composition of q skew derivations in banach algebras abstract let a be a banach algebra let be an automorphism of a and let d be q skew derivations of a we show that if d a is quasi nilpotent for any a a then d a 3 lies in the radical of a for all a a this result simultaneously generalizes the previous results obtained by bresar and semrl 5 chebotar et al 7 and abdelali 1

prevalence a translation invariant almost every on infinite dimensional spaces we present a measure theoretic condition for a property to hold almost everywhere on an infinite dimensional vector space with particularemphasis on function spaces such as c k and l p like the concept of lebesgue almost every on an infinite dimensional spaces our notion of prevalence is translation invariant instead of using a specific measure on the entire space we define prevalence in terms of the class of all probability measures with compact support prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space

restricted markov uniqueness for the stochastic quantization of p φ 2 and its applications abstract in this paper we obtain restricted markov uniqueness of the generator and uniqueness of martingale probabilistically weak solutions for the stochastic quantization problem in both the finite and infinite volume case by clarifying the precise relation between the solutions to the stochastic quantization problem obtained by the dirichlet form approach and those obtained in 10 and in 24 we prove that the solution x z where x is obtained by the dirichlet form approach in 4 and z is the corresponding o u process satisfies the corresponding shifted equation see 1 4 below moreover we obtain that the infinite volume p 2 quantum field is an invariant measure for the x y z where y is the unique solution to the shifted equation

affiliated series and extensions of modules the main purpose of this paper is to examine the influence on the representation theory of a noetherian ring of the prime ideal structure of the ring the results concern the affiliated primes of a module defined below we show how the affiliated primes are connected with the annihilator primes and we prove a uniqueness theorem for affiliated primes under certain conditions we then proceed to analyse what affiliated primes can appear in terms of the ideal structure of the ring with special reference to the fundamental primes of injective modules our starting point is a theorem of jategaonkar which analyses the structure of modules with exactly two affiliated primes and which already has consequences for uniserial modules of length two we give a generalization of this result for modules with longer affiliated series and in particular for uniserial modules of length greater than two in particular we show that some of the possibilities one might have envisioned in the light of jategaonkar s theorem cannot in fact occur and we demonstate a somewhat unexpected connection between affiliated primes and annihilator primes to put all this material in context we begin section one with an analysis of what it all means for uniserial modules of length three we then turn to the uniqueness question for affiliated primes here and in the rest of the paper we find it desirable to have some kind of dimension 164 0021 8693 91 3 00

examples of ore extensions which are maximal orders whose based rings are not maximal orders abstract let r be a prime goldie ring and be a skew derivation on r it is well known that if r is a maximal order then the ore extension r x is a maximal order it was a long standing open question that the converse is true or not in case 1 and 0 we give an example of non maximal order r with a skew derivation on r 1 0 such that r x is a maximal order

elliptic equations for measures on infinite dimensional spaces and applications we introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces this concept allows one to consider equations whose coefficients are not globally integrable by using a suitably extended lyapunov function technique we derive a priori estimates for the solutions of such equations and prove new existence results as an application we consider stochastic burgers reaction diffusion and navier stokes equations and investigate the elliptic equations for the corresponding invariant measures our general theorems yield a priori estimates and existence results for such elliptic equations we also obtain moment estimates for gibbs distributions and prove an existence result applicable to a wide class of models

holder continuity for the inverse of mane s projection abstract in a celebrated work mane showed that given a compact set x with fractal dimension d in a banach space b there exists a projection p of rank 2 d 1 such that p restricted to x is injective here we prove a stronger result when b is finite dimensional namely that there exists an orthogonal projection p 0 such that not only p 0 is injective on x but also its inverse is holder continuous when restricted to p 0 x

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

uniqueness of generalized schrodinger operators part ii abstract we prove that for h 1 2 loc r d dx 0 dx a e the generalized schrodinger operator s 2 1 dom s c 0 r d has exactly one self adjoint extension on l 2 r d 2 dx which generates a sub markovian semigroup on l 2 r d 2 dx this is based on our previous work where a necessary and sufficient condition on for this to hold was proved but which was only verified to always hold if d 1 we also prove a corresponding result where r d is replaced by an infinite dimensional space and the lebesgue measure by some gaussian measure whose covariance operator has a discrete spectrum

links between prime ideals in differential operator rings in recent years much work has been done on a localization theory for noetherian rings links between prime ideals play a central role in this development when a ring r is not commutative it is rarely possible to localize r at a prime ideal p having a link between p and another prime ideal is an obstruction to localization in this paper we will investigate the links between prime ideals in differential operator rings showing that it is possible to localize these rings in a certain sense recall that a multiplicatively closed subset s of a ring r is a right ore set if for all r in r and all s in s we have rs n sr qr the set s is an ore set if it is both a right and a left ore set it is always possible to construct a ring rf where the elements of s become units however if we insist on writing the elements of rf as right fractions rs with r in r and s in s then the set s must be a right ore set 9 sect 12 11 let p be a prime ideal of r and let 7 p r e r 1 r p is regular in r p we say that p is a localizable prime ideal if the set u p is an ore set we will shortly give an example of a prime ideal which is not localizable but first we introduce the differential operator ring let r be a ring and let 6 be an additive map from r to itself we call 6 a derivation if it satisfies the product rule 6 ab a b ad b the differential operatqr ring over r denoted r 0 s is a free left r module with basis 1 8 13 thus every element of r 0 s is a polynomial in 0 with coefficients from r the addition in r l s is defined as usual for polynomials but multiplication is extended from r by the rule br rti 6 r the differential operator ring is an associative ring and is determined up to isomorphism by an obvious universal property 9 sect 12 21

stochastic analysis on product manifolds dirichlet operators on differential forms abstract we define a de rham complex over a product manifold infinite product of compact manifolds and dirichlet operators on differential forms associated with differentiable measures in particular gibbs measures which generalize the notions of bochner and de rham laplacians we give probabilistic representations for corresponding semigroups and study properties of the corresponding stochastic dynamics

stochastic processes associated with kms states abstract a general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes stochastically positive kms systems periodic os positive stochastic processes periodic positive semigroup structures are defined and shown to be equivalent the objects of the tomita takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven a two sided markov property for the stochastic process is related to a cyclicity property of the kms state perturbations are constructed by the feynman kac nelson formula the general framework is applied to kms states given by density matrices

stochastic quantization of field theory in finite and infinite volume abstract using the theory of dirichlet forms a distribution valued symmetric markov process is constructed so that it has the 4 2 measure as its invariant probability measure several properties of this process are proved and a finite to infinite volume limit theorem is established

uniqueness of gibbs states for quantum lattice systems we prove uniqueness of euclidean gibbs states for certain quantum lattice systems with unbounded spins we use dobrushin s uniqueness criterion the necessary estimates for the vasershtein distance between the corresponding one point conditional distributions with boundary conditions differing only at one side are obtained by proving a log sobolev inequality on the infinite dimensional single spin loop spaces some important classes of concrete examples to which all this applies are discussed

uniqueness and the global markov property for euclidean fields the case of general exponential interaction the uniqueness and the global markov property for the regular gibbs measure corresponding to the interaction u lambda varphi lambda int limits lambda d 2 x int d varrho alpha e alpha varphi 0 x for 0 d a probability measure with support in 2 sqrt pi 2 sqrt pi is proved

orthogonal sequences and regularity of embeddings into finite dimensional spaces abstract this paper focuses on the regularity of linear embeddings of finite dimensional subsets of hilbert and banach spaces into euclidean spaces we study orthogonal sequences in a hilbert space h whose elements tend to zero and similar sequences in the space c 0 of null sequences the examples studied show that the results due to hunt and kaloshin regularity of embeddings of infinite dimensional fractal sets into finite dimensional spaces nonlinearity 12 1999 1263 1275 and robinson linear embeddings of finite dimensional subsets of banach spaces into euclidean spaces nonlinearity 22 2009 711 728 for subsets of hilbert and banach spaces with finite box counting dimension are asymptotically sharp an analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by olson and robinson almost bi lipschitz embeddings and almost homogeneous sets trans amer math soc 362 1 2010 145 168 for subsets x of hilbert spaces when x x has finite assouad dimension

uniqueness of gibbs states for general p ϕ 2 weak coupling models by cluster expansion we consider quantum fields with weak coupling in two space time dimensions we prove that the set of their ultraregular gibbs states consists of only one point and this point is an extremal gibbs state

almost bi lipschitz embeddings and almost homogeneous sets this paper is concerned with embeddings of homogeneous spaces into euclidean spaces we show that any homogeneous metric space can be em bedded into a hilbert space using an almost bi lipschitz mapping bi lipschitz to within logarithmic corrections the image of this set is no longer homo geneous but almost homogeneous we therefore study the problem of em bedding an almost homogeneous subset x of a hilbert space h into a finite dimensional euclidean space in fact we show that if x is a compact subset of a banach space and x x is almost homogeneous then for n sufficiently large a prevalent set of linear maps from x into r n are almost bi lipschitz between x and its image we are then able to use the kuratowski embedding of x d into l x to prove a similar result for compact metric spaces

prime ideals in skew polynomial rings and quantized weyl algebras the concern of this paper is to investigate the structure of skew polynomial rings ore extensions of the form t r where and are both nontrivial and in particular to analyze the prime ideals of t the main focus is on the case that r is commutative noetherian in this case the prime ideals of t are classified polynomial identities and artin rees separation in prime factor rings are investigated and cliques of prime ideals are studied the second layer condition is proved as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique futher q skew derivations on noncommutative coefficient rings are introduced and some preliminary results on contractions of prime ideals of t are obtained in this setting finally prime ideals in quantized weyl algebras over fields are analyzed

ergodicity for the stochastic dynamics of quasi invariant measures with applications to gibbs states the convex set maof quasi invariant measures on a locally convex spaceewith given shift radon nikodym derivatives i e cocycles a atk k k0 t ris analyzed the extreme points of maare characterized and proved to be non empty a specification of lattice type is constructed so that macoincides with the set of the corresponding gibbs states as a consequence via a well known method due to dynkin and follmer a unique representation of an arbitrary element in main terms of extreme ones is derived furthermore the corresponding classical dirichlet forms e d e and their associated semigroups t t t 0onl2 e are discussed under a mild positivity condition it is shown that mais extreme if and only if e d e is irreducible or equivalently t t t 0is ergodic this implies time ergodicity of associated diffusions applications to gibbs states of classical and quantum lattice models as well as those occuring in euclidean quantum field theory are presented in particular it is proved that the stochastic quantization of a guerra rosen simon gibbs state on d r2 ininfinite volumewith polynomial interaction is ergodic if the gibbs state is extreme i e is a pure phase which solves a long standing open problem

the extended centroid and x inner automorphisms of ore extensions for the ore extension r t s d where r is a prime ring we determine the center the extended centroid and the x inner automorphisms the results depend on the structure of ideals of r t s d

euclidean gibbs measures of interacting quantum anharmonic oscillators a rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting dimensional quantum anharmonic oscillators is given the oscillators are indexed by the elements of a countable set mathbb l subset mathbb r d possibly irregular the anharmonic potentials vary from site to site and the interaction has infinite range the description is based on the representation of the gibbs states in terms of path measures the so called euclidean gibbs measures it is proven that a the set of such measures mathcal g rm t is non void and compact b every mu in mathcal g rm t obeys an exponential integrability estimate the same for the whole set mathcal g rm t c every mu in mathcal g rm t has a lebowitz presutti type support d mathcal g rm t is a singleton at high temperatures the case of attractive interaction and nu 1 is studied in more detail we prove that a mathcal g rm t 1 at low temperatures b mathcal g rm t 1 due to quantum effects and at a nonzero external field thereby a qualitative theory of phase transitions and quantum effects which interprets most important experimental data known for the corresponding physical objects is developed

specifications and martin boundaries for scr p phi _2 random fields it is shown thatp 2 gibbs states in the sense of guerra rosen and simon are given by a specification the construction of the specification is based on finding a proper version of the interaction density given by the polynomialp the existence of this version follows from the fact that all powers of the solution of a dirichlet problem for an open bounded setu with boundary data given by a distribution are integrable onu as a consequence the martin boundary theory for specifications can be applied top 2 random fields it follows that anyp 2 gibbs state can be represented in terms of extreme gibbs states in certain cases the extreme gibbs states are characterized in terms of harmonic functions it follows in particular that for any given boundary condition introduced so far the associated cutoffp 2 measure has a representation as an integral over harmonic functions

algebraic q skew derivations abstract let r be a prime ring c its extended centroid and r f resp q its left resp symmetric martindale quotient ring let be a derivation of r where is an automorphism of r we show the equivalence of k polynomials resp k identities of and cv polynomials resp semi invariant polynomials in the ore extension q x we prove the existence of k polynomials of in certain rather general family of maps as applications the following are proved among other things consider the expression x i 0 n a i i x where a i r f and a n 0 1 if dim c r c then either r is a gpi ring or x 0 for all x r f 2 if r c then either r is commutative or x 0 for all x r f

dirichlet operators via stochastic analysis abstract we give sufficient conditions for essential self adjointness of symmetric differential operators dirichlet operators associated with classical dirichlet forms given by probability measures on hilbert spaces under an additional assumption of log concavity for the measures we prove the existence of a gap at the lower end of spectrum of the dirichlet operator and the ergodicity of the corresponding semi group we consider also applications to the study of stochastic dynamics in some classical lattice systems