codimensions of algebras and growth functions abstract let a be an algebra over a field f of characteristic zero and let c n a n 1 2 be its sequence of codimensions we prove that if c n a is exponentially bounded its exponential growth can be any real number 1 this is achieved by constructing for any real number 1 an f algebra a such that lim n c n a n exists and equals the methods are based on the representation theory of the symmetric group and on properties of infinite sturmian and periodic words

gradient estimate for solutions to poisson equations in metric measure spaces abstract let x d be a complete pathwise connected metric measure space with a locally ahlfors q regular measure where q 1 suppose that x d supports a local 1 2 poincare inequality and a suitable curvature lower bound for the poisson equation u f on x d moser trudinger and sobolev inequalities are established for the gradient of u the local holder continuity with optimal exponent of solutions is obtained

tensor product theorems in positive characteristic abstract in this paper we study tensor products of t prime t ideals over infinite fields the behaviour of these tensor products over a field of characteristic 0 was described by kemer first we show using methods due to regev that such a description holds if one restricts oneself to multilinear polynomials only second applying graded polynomial identities we prove that the tensor product theorem fails for the t ideals of the algebras m 1 1 e and e e where e is the infinite dimensional grassmann algebra m 1 1 e consists of the 2 2 matrices over e having even i e central elements of e and the other diagonal consisting of odd anticommuting elements of e note that these proofs do not depend on the structure theory of t ideals but are elementary ones all this comes to show once more that the structure theory of t ideals is essentially about the multilinear polynomial identities

t ideals and superazumaya algebras equivalent to being azumaya of finite rank thus one would expectazumaya algebras to play a role in the theory of t prime algebras a pointof view already taken in 2 wxour goal in this paper is to link the theory of t prime algebras to theseother important theories the theory of zr2z graded polynomial identi ties and the graded theory of azumaya algebras following kemer we believe that there is a natural connection between pi theory and zr2z graded algebras and that pi theorems should tend to have natural zr2z graded analogues our main result will be such an analogue of the artin procesi theorem theorems 13 18 in the process we consider the brauer wall group defined by c t c wall in 17 and further developed by h bass in 1 and c small in 16 cf wx wx wxwx8 for further results we will construct a pi criterion for a zr2z gradedalgebra to be zr2z graded projective of constant rank throughout thispaper all algebras will be with 1 over an infinite field

group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

representability and specht problem for g graded algebras abstract let w be an associative pi algebra over a field f of characteristic zero graded by a finite group g let id g w denote the t ideal of g graded identities of w we prove 1 g graded pi equivalence there exists a field extension k of f and a finite dimensional z 2 z g graded algebra a over k such that id g w id g a where a is the grassmann envelope of a 2 g graded specht problem the t ideal id g w is finitely generated as a t ideal 3 g graded pi equivalence for affine algebras let w be a g graded affine algebra over f then there exists a field extension k of f and a finite dimensional algebra a over k such that id g w id g a

on the polynomial identities of the algebra m11 e abstract verbally prime algebras are important in pi theory they were described by kemer over a field k of characteristic zero 0 and k t the trivial ones m n k m n e m ab e here k t is the free associative algebra of infinite rank with free generators t e denotes the infinite dimensional grassmann algebra over k m n k and m n e are the n n matrices over k and over e respectively the algebras m ab e are subalgebras of m a b e see their definition below the generic also called relatively free algebras of these algebras have been studied extensively procesi described the generic algebra of m n k and lots of its properties models for the generic algebras of m n e and m ab e are also known but their structure remains quite unclear in this paper we study the generic algebra of m 11 e in two generators over a field of characteristic 0 in an earlier paper we proved that its centre is a direct sum of the field and a nilpotent ideal of the generic algebra and we gave a detailed description of this centre those results were obtained assuming the base field infinite and of characteristic different from 2 in this paper we study the polynomial identities satisfied by this generic algebra we exhibit a basis of its polynomial identities it turns out that this algebra is pi equivalent to a 5 dimensional algebra of certain upper triangular matrices the identities of the latter algebra have been studied these were described by gordienko as an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of m 11 e also we describe the polynomial identities in two variables of the algebra m 11 e

graded identities of matrix algebras and the universal graded algebra in the last decade group gradings and graded identities of finite dimensional central simple algebras have been an active area of research we refer the reader to bahturin et al 6 and 7 there are two basic kinds of group grading elementary and fine it was proved by bahturin and zaicev 7 that any group grading of mn c is given by a certain composition of an elementary grading and a fine grading in this paper we are concerned with fine gradings on mn c and their corresponding graded identities let r be a simple algebra finite dimensional over its center k and g a finite group we say that r is fine graded by g if r g grg is a grading and dimk rg 1 thus any component is either 0 or isomorphic to k as a k vector space it is easy to show that supp r the subset of elements of g for which rg is not 0 is a subgroup of g moreover

poisson equation poincare lelong equation and curvature decay on complete kahler manifolds in the first part of this work the poisson equation on complete noncompact manifolds with nonnegative ricci curvature is studied sufficient and necessary conditions for the existence of solutions with certain growth rates are obtained sharp estimates on the solutions are also derived in the second part these results are applied to the study of curvature decay on complete kahler manifolds in particular the poincare lelong equation on complete noncompact kahler manifolds with nonnegative holomorphic bisectional curvature is studied several applications are then derived which include the steinness of the complete kahler manifolds with nonnegative curvature and the flatness of a class of complete kahler manifolds satisfying a curvature pinching condition liouville type results for plurisubharmonic functions are also obtained 0 introduction in this paper we will discuss the poisson equation on complete noncompact manifolds and derive some applications on kahler manifolds let mm be a complete noncompact kahler manifold where m 2 is the complex dimension assume m has nonnegative holomorphic bisectional curvature and has maximal volume growth such that the scalar curvature decays like r 2 where r is the distance from a fixed point then it was proved in 20 by mok siu yau that one can solve the following poincare lelong equation 0 1 1 u received november 25 2000 and in revised form june 28 2001 the first author was partially supported by nsf grant dms9970284 usa the second author was partially supported by nsf of china project 10001001 and the third author was partially supported by earmarked grant of hong kong cuhk4217 99p

gradient estimates for heat kernels and harmonic functions abstract let x d be a doubling metric measure space endowed with a dirichlet form e deriving from a carre du champ assume that x d e supports a scale invariant l 2 poincare inequality in this article we study the following properties of harmonic functions heat kernels and riesz transforms for p 2 i g p l p estimate for the gradient of the associated heat semigroup ii r h p l p reverse holder inequality for the gradients of harmonic functions iii r p l p boundedness of the riesz transform p iv g b e a generalised bakry emery condition we show that for p 2 i ii iii are equivalent while for p i ii iv are equivalent moreover some of these equivalences still hold under weaker conditions than the l 2 poincare inequality our result gives a characterisation of li yau s gradient estimate of heat kernels for p while for p 2 it is a substantial improvement as well as a generalisation of earlier results by auscher coulhon duong hofmann 7 and auscher coulhon 6 applications to isoperimetric inequalities and sobolev inequalities are given our results apply to riemannian and sub riemannian manifolds as well as to non smooth spaces and to degenerate elliptic parabolic equations in these settings

trace identities and 2 graded invariants we prove razmyslov s theorem on trace identities for mk i using the invariant theory of pl k i in 5 and 6 razmyslov and procesi independently proved their celebrated theorem on trace identities of the algebra of n x n matrices their work completely characterized these trace identities both in the sense of showing that they all result from the cayley hamilton theorem and in the sense of calculating their cocharacter recently razmyslov 7 generalized these results to describe the trace identities of mk l a z 2z graded analogue of the matrix algebra in the present paper we reprove razmyslov s results on mk l using methods based on procesi s this not only gives new insights into razmyslov s theorem but also entails the development of some z 2z graded invariant theory which we believe to be of interest in its own right we now define our basic objects and state our main theorems throughout this paper we will be working over a field f of characteristic zero e will denote an infinite dimensional grassmann algebra over f the algebra e has a natural z 2z grading with respect to which it is a graded commutative algebra the degree zero part eo is spanned by all words in e of even length and the degree one part e1 is spanned by all words of odd length then the statement that e is graded commutative means that if a b e e are homogeneous then ab l dega degbba we will add and multiply degrees as elements of z 2z we will denote by v the free e module on the basis tl tk 1ll 1ll the module v has a z 2z grading gotten by declaring the t s to be degree zero and the s to be degree one there are three sets of maps v v which play important roles in this paper one is the ring of e endomorphisms of v ende v if we use our basis tl tk 1ll 1ll to identify elements of v with column vectors with entries in e then elements of ende v will be identified with k i x k 1 matrices with entries in e acting on the left of v to define ende v in a basis free manner we first make v into a left e module by setting ve l degt degeev for all homogeneous v e v e e e and then extend to all of e and v by linearity then ende v can be defined as the set of all additive maps f v v which commute with the left action of e another set of maps which we will make use of is pl v the general linear lie supergroup and it is a subgroup of ende v we define pl v as f e ende v is invertible and for all homogeneous v e v degf v degv as k l x k l matrices elements of pl v look like c d x where a is a k x k matrix with entries in eo d is an i x i matrix with entries in eo and b and c have entries in e1 received by the editors june 24 1987 1980 mathematics subject classification 1985 revision primary 16a38 partially supported by nsf grant dms 870 3481 g31988 american mathematical society 0002 9947 88 1 00 25 per page

gradient estimates for variable coefficient parabolic equations and singular perturbation problems in this article we prove via monotonicity formulas interior and boundary gradient esti mates for solutions to second order parabolic equations in divergence form with dini top order coefficients we then prove uniform lipschitz estimates for solutions of singular perturbation prob lems using the previous results and two phase monotonicity formulas introduction the purpose of this paper is to study the regularity of spatial gradients of solutions to second order uniformly parabolic equations in divergence form with bounded lower order terms and dini continuous coefficients and also to establish uniform spatial lipschitz estimates for some singular perturbation problems in the same context in part 1 we obtain interior and boundary estimates for solutions to these linear equations as is well known the weakest assumption on the modulus of continuity of the top order terms to obtain interior continuity of the gradient of solutions is dini continuity see part 1 for the relevant definitions see for instance g w in the elliptic case the usual approach to such results is to use appropriate potentials here we use a monotonicity formula approach theo rem 1 2 9 which we believe is new the approach has the advantage that it allows us to show that dini continuity of the coefficients at a given point yields bound edness of the gradient at the same point using our monotonicity formula the fact that linear functions are almost solutions and a geometric argument from cl we also obtain under uniform dini continuity of the coefficients a uniform modulus of continuity estimate for gradients of solutions theorem 1 3 1 the fact that the monotonicity formula at a point depends only on the dini continuity at that same point enables us to use a simple reflection argument to extend these results up to the boundary theorem 1 4 5 and theorem 1 4 12 we thus obtain a modulus of continuity estimate up to the boundary for gradients of solutions with smooth dirichlet data under uniform dini continuity of the top order coeffi cients once this is established a compactness argument theorem 1 5 10 and a judicious application of harnack s principle allow us to obtain a hopf maximum principle theorem 1 5 10 corollary 1 5 13 and corollary 1 5 16 for our class of equations

tensor products of matrix algebras over the grassmann algebra abstract we develop elementary tools for studying p i equivalences between algebras which are tensor products of matrix subalgebras we then deduce kemer s results about the p i equivalence of tensor products of matrix subalgebras over the grassmann algebra

the centre of generic algebras of small pi algebras abstract verbally prime algebras are important in pi theory they are well known over a field k of characteristic zero 0 and k t the trivial ones m n k m n e m a b e here k t is the free associative algebra with free generators t e is the infinite dimensional grassmann algebra over k m n k and m n e are the n n matrices over k and over e respectively moreover m a b e are certain subalgebras of m a b e defined below the generic algebras of these algebras have been studied extensively procesi gave a very tight description of the generic algebra of m n k the situation is rather unclear for the remaining nontrivial verbally prime algebras in this paper we study the centre of the generic algebra of m 11 e in two generators we prove that this centre is a direct sum of the field and a nilpotent ideal of the generic algebra we describe the centre of this algebra as a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by berele

multialternating jordan polynomials and codimension growth of matrix algebras abstract let r be the jordan algebra of k k matrices over a field of characteristic zero we exhibit a noncommutative jordan polynomial f multialternating on disjoint sets of variables of order k 2 and we prove that f is not a polynomial identity of r we then study the growth of the polynomial identities of the jordan algebra r through an analysis of its sequence of jordan codimensions by exploiting the basic properties of the polynomial f we are able to prove that the exponential rate of growth of the sequence of jordan codimensions of r in precisely k 2

graded polynomial identities for matrices with the transpose involution abstract let g be a group of order k we consider the algebra m k c of k by k matrices over the complex numbers and view it as a crossed product with respect to g by embedding g in the symmetric group s k via the regular representation and embedding s k in m k c in the usual way this induces a natural g grading on m k c which we call a crossed product grading we study the graded identities for m k c equipped with such a crossed product grading and the transpose involution to each multilinear monomial in the free graded algebra with involution we associate a directed labeled graph use of these graphs allows us to produce a set of generators for the t ideal of identities it also leads to new proofs of the results of kostant and rowen on the standard identities satisfied by skew matrices finally we determine an asymptotic formula for the graded codimension of m k c

elementary gradings on the lie algebra utn the algebras utn k utn k of the upper triangular matrices over a field k are of significant importance in the theory of algebras with polynomial identities group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question in this paper we consider the lie algebra utn k utn k of all upper triangular matrices of order n we study the group gradings on this algebra it turns out that the gradings on the lie algebra utn k utn k are much more intricate than those in the associative case in this paper we describe the elementary gradings on the lie algebra utn k utn k finally we study the canonical grading on utn k utn k by the cyclic group znzn of order n we produce a finite basis of the graded polynomial identities satisfied by this grading

basis of the identities of the matrix algebra of order two over a field of characteristic p 2 abstract in this paper we prove that the polynomial identities of the matrix algebra of order 2 over an infinite field of characteristic p 2 admit a finite basis we exhibit a finite basis consisting of four identities and in almost all cases for p we describe a minimal basis consisting of two identities the only possibilities for p where we do not exhibit minimal bases of these identities are p 3 and p 5 we show that when p 3 one needs at least three identities and we conjecture a minimal basis in this case in the course of the proof we construct an explicit basis of the vector space of the central commutator polynomials modulo the ideal of the identities of the matrix algebra of order two

critical end point behavior in a binary fluid mixture we consider the liquid gas phase boundary in a binary fluid mixture near its critical end point using general scaling arguments we show that the diameter of the liquid gas coexistence curve exhibits singular behaviour as the critical end point is approached this prediction is tested by means of extensive monte carlo simulations of a symmetrical lennard jones binary mixture within the grand canonical ensemble the simulation results show clear evidence for the proposed singularity as well as confirming a previously predicted singularity in the coexistence chemical potential fisher and upton phys rev lett 65 2402 1990 the results suggest that the observed singularities particularly that in the coexistence diameter should also be detectable experimentally

the poincare inequality is an open ended condition let p 1 and let x d be a complete metric measure space with borel and doubling that admits a 1 p poincare inequality then there exists e 0 such that x d admits a 1 q poincare inequality for every q p e quantitatively

geometry and analysis of dirichlet forms let e be a regular strongly local dirichlet form on l2 x m and d the associated intrinsic distance assume that the topology induced by d coincides with the original topology on x and that x is compact satisfies a doubling property and supports a weak 1 2 poincare inequality we first discuss the non coincidence of the intrinsic length structure and the gradient structure under the further assumption that the ricci curvature of x is bounded from below in the sense of lott sturm villani the following are shown to be equivalent i the heat flow of e gives the unique gradient flow of u ii e satisfies the newtonian property iii the intrinsic length structure coincides with the gradient structure moreover for the standard resistance dirichlet form on the sierpinski gasket equipped with the kusuoka measure we identify the intrinsic length structure with the measurable riemannian and the gradient structures we also apply the above results to the coarse ricci curvatures and asymptotics of the gradient of the heat kernel

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

lipschitz continuity of cheeger harmonic functions in metric measure spaces abstract we use the heat equation to establish the lipschitz continuity of cheeger harmonic functions in certain metric spaces the metric spaces under consideration are those that are endowed with a doubling measure supporting a 1 2 poincare inequality and in addition supporting a corresponding sobolev poincare type inequality for the modification of the measure obtained via the heat kernel examples are given to illustrate the necessity of our assumptions on these spaces we also provide an example to show that in the general setting the best possible regularity for the cheeger harmonic functions is lipschitz continuity

graded codimensions of lie superalgebra b 2 abstract we study asymptotic behaviour of graded codimensions of lie superalgebra b 2 we prove that graded pi exponent exists and is equal to 3 2 3

volume growth and spectrum for general graph laplacians we prove estimates relating exponential or sub exponential volume growth of weighted graphs to the bottom of the essential spectrum for general graph laplacians the volume growth is computed with respect to a metric adapted to the laplacian and use of these metrics produces better results than those obtained from consideration of the graph metric only conditions for absence of the essential spectrum are also discussed these estimates are shown to be optimal or near optimal in certain settings and apply even if the laplacian under consideration is an unbounded operator

polynomial identities of algebras in positive characteristic the verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic p 2 little is known about them in previous papers we discussed some sharp differences between these two cases for the characteristic and we showed that the so called tensor product theorem is in part no longer valid in the second case in this paper we study the gelfand kirillov dimension of the relatively free algebras of verbally prime and related algebras we compute the gk dimensions of several algebras and thus obtain a new proof of the fact that the algebras m1 1 e and e e are not pi equivalent in characteristic p 2 furthermore we show that the following algebras are not pi equivalent in positive characteristic ma b e e and ma b e ma b e e and mc d e e when a b c d a b

logarithmic sobolev inequalities on noncompact riemannian manifolds this paper presents a dimension free harnack type inequality for heat semigroups on manifolds from which a dimension free lower bound is obtained for the logarithmic sobolev constant on compact manifolds and a new criterion is proved for the logarithmic sobolev inequalities abbrev lsi on noncompact manifolds as a result it is shown that lsi may hold even though the curvature of the operator is negative everywhere

identities and isomorphisms of graded simple algebras abstract let g be an arbitrary abelian group and let a and b be two finite dimensional g graded simple algebras over an algebraically closed field f such that the orders of all finite subgroups of g are invertible in f we prove that a and b are isomorphic if and only if they satisfy the same g graded identities we also describe all isomorphism classes of finite dimensional g graded simple algebras

polynomial codimension growth and the specht problem abstract we construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4 this is achieved by building for any real number 0 1 a commutative nonassociative algebra a whose codimension sequence c n a n 1 2 is polynomially bounded and lim log n c n a 3 as an application we are able to construct a new example of a variety with an infinite basis of identities

invariant theory for matrices over the grassmann algebra we prove that the algebra of polynomial maps mn e e invariant under simultaneous conjugation is generated by the queer trace polynomials

graded identities of simple real graded division algebras abstract let a and b be finite dimensional simple real algebras with division gradings by an abelian group g in this paper we give necessary and sufficient conditions for the coincidence of the graded identities of a and b we also prove that every finite dimensional simple real algebra with a g grading satisfies the same graded identities as a matrix algebra over an algebra d with a division grading that is either a regular grading or a non regular pauli grading moreover we determine when the graded identities of two such algebras coincide for graded simple algebras over an algebraically closed field it is known that two algebras satisfy the same graded identities if and only if they are isomorphic as graded algebras

field theoretical analysis of singularities at critical end points abstract continuum models with critical end points are considered whose hamiltonian h depends on two densities and field theoretic methods are used to show the equivalence of the critical behavior on the critical line and at the critical end point and to give a systematic derivation of critical end point singularities like the thermal singularity t 2 of the spectator phase boundary and the coexistence singularities t 1 or t of the secondary density the appearance of a discontinuity eigenexponent associated with the critical end point is confirmed and the mechanism by which it arises in field theory is clarified

colength sequences for kemer algebras this paper is a sequel to 4 in that paper we calculated the asymptotic behavior of the colength sequence for k k matrices and in this paper we will calculate the asymptotic behavior of the colength sequence for the algebras mk given a p i algebra a the cocharacter sequence n a n 0 is a sequence of sncharacters if we decompose each character into a sum of irreducible sn characters we would get something of the form n a m a let ln a be the length of n a so ln a par n m a the sequence of numbers ln a is the colength sequence of a similar constructions can also be made for other cocharacter sequences such as the trace cocharacter if a is an algebra with trace kemer classified verbally prime p i algebras into three families see 8 one is k k matrices whose colength sequence we investigated in 4 one is k k matrices over an infinite dimensional grassmann algebra we do not know yet how to deal with this case and the third is the algebras mk let e be an infinite dimensional grassmann algebra and let k n then mk is a subalgebra of the matrix algebra mn e in order to define it note that the grassmann algebra has a natural z 2z grading e e0 e1 with respect to which it is anticommutative i e degree 0 elements are central and degree 1 elements anticommute with each other let i and j be between 1 and n we define the ordered pair i j to be of degree 0 if either 1 i j k or k 1 i j n and of degree 1 otherwise then mk will be the set of all matrices aij mn e in which the degree of each aij

on the graded identities for elementary gradings in matrix algebras over infinite fields we find a basis for the g graded identities of the n n matrix algebra mn k over an infinite field k of characteristic p 0 with an elementary grading such that the neutral component corresponds to the diagonal of mn k

colength sequences for matrices over grassmann algebras abstract we prove that the colength sequence of k k matrices over the grassmann algebra is asymptotic to a constant times n k 4

critical endpoints in the spherical model with rapidly decaying power law interactions abstract we study the d dimensional spherical model with a rapidly decaying power law interaction j r 1 r d with 2 d 2 3 introducing a potential w s ds 2 u s 4 vs 6 at each site the phase diagrams for different values of d and are shown to exhibit nonclassical critical lines terminating at critical endpoints we show that although these lines exhibit the usual spherical model leading order critical behavior independent of the specific range of significant dependent correction to scaling terms are present when the interaction decays fast with 2

graded polynomial identities and codimensions computing the exponential growth abstract let g be a finite abelian group and a a g graded algebra over a field of characteristic zero this paper is devoted to a quantitative study of the graded polynomial identities satisfied by a we study the asymptotic behavior of c n g a n 1 2 the sequence of graded codimensions of a and we prove that if a satisfies an ordinary polynomial identity lim n c n g a n exists and is an integer we give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple g z 2 graded algebra related to a

central polynomials in the matrix algebra of order two abstract we exhibit minimal bases of the polynomial identities for the matrix algebra m 2 k of order two over an infinite field k of characteristic p 2 we show that when p 3 the t ideal of this algebra is generated by three independent identities and when p 3 one needs only two identities the standard identity of degree four and the hall identity note that the same holds when the base field is of characteristic 0 furthermore using the exact form of the basis of the identities for m 2 k we give finite minimal set of generators of the t space of the central polynomials for the algebra m 2 k the set of generators depends on the characteristic of the field as well

isoperimetric inequality from the poisson equation via curvature in this paper we establish an isoperimetric inequality in a metric measure space via the poisson equation let x d be a complete pathwise connected metric space with locally ahlfors q regular measure where q 1 that supports a local l2 poincare inequality we show that for the poisson equation u g if the local l norm of the gradient du can be bounded by the lorentz norm lq 1 of g then we obtain an isoperimetric inequality and a sobolev inequality in x d with optimal exponents by assuming a suitable curvature lower bound we establish such optimal bounds on 2011 wiley periodicals inc

group graded algebras and almost polynomial growth let f be a field of characteristic 0 g a finite abelian group and a a g graded algebra we prove that a generates a variety of g graded algebras of almost polynomial growth if and only if a has the same graded identities as one of the following algebras 1 fcp the group algebra of a cyclic group of order p where p is a prime number and p g 2 ut2g f the algebra of 2 2 upper triangular matrices over f endowed with an elementary g grading 3 e the infinite dimensional grassmann algebra with trivial g grading 4 in case 2 g ez2 the grassmann algebra with canonical z2 grading

identities of bilinear mappings and graded polynomial identities of matrices we describe the polynomial identities of the bilinear mappings v w k and vr v r mr k where v w vr are finite dimensional vector spaces over a field k of characteristic 0 dim vr r and mr k is the r r matrix algebra we show that these identities follow from the commutativity of the values in k of the bilinear forms v w k and v r vr k and capelli identities we apply these results to the g graded polynomial identities of the algebra mr k with the elementary g grading associated with the rtuple g g h where g is an arbitrary group and g h g are such that g 1 h 2 e i n

poincare inequality from remote balls to all balls abstract it is shown that the poincare inequality holds uniformly for all balls whenever it holds for z remote balls providing the set z satisfies some additional conditions including the condition that z does not separate the space the aim of this paper is to prove similar results without using this assumption

graded identities for tensor products of matrix super algebras over the grassmann algebra abstract in this paper we study the graded identities satisfied by the superalgebras m a b over the grassmann algebra and by their tensor products these algebras play a crucial role in the theory developed by a kemer that led to the solution of the long standing specht problem it is well known that over a field of characteristic 0 the algebras m pr qs ps qr and m p q m r s satisfy the same ordinary polynomial identities by means of describing the corresponding graded identities we prove that the t ideal of the former algebra is contained in the t ideal of the latter furthermore the inclusion is proper at least in case r s 1 1 finally we deal with the graded identities satisfied by algebras of type m 2 n 1 2 n 1 and relate these graded identities to the ones of tensor powers of the grassmann algebra our proofs are combinatorial and rely on the relationship between graded and ordinary identities as well as on appropriate models for the corresponding relatively free graded algebras

modulus and the poincare inequality on metric measure spaces the purpose of this paper is to develop the understanding of modulus and the poincare inequality as defined on metric measure spaces various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces consequently modulus is shown to be upper semi continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces moreover several competing definitions for the poincare inequality are shown to coincide if the underlying measure is doubling one such characterization considers only continuous functions and their continuous upper gradients and extends work of heinonen and koskela applications include showing that the p poincare inequality with a doubling measure for p 1 persists through to the limit of a sequence of converging pointed metric measure spaces this extends results of cheeger a further application is the construction of new doubling measures in euclidean space which admit a 1 poincare inequality

group gradings on the lie algebra of upper triangular matrices abstract the algebras u t n of the n n upper triangular matrices over a field k are of significant importance in the theory of algebras with polynomial identities group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question we classify the group gradings on the lie algebra u t n it was proved by valenti and zaicev in 2007 that every group grading on the associative algebra u t n is isomorphic to an elementary grading the elementary gradings on u t n are also well understood see 6 it follows from our results that there are nonelementary gradings on u t n thus the gradings on the lie algebra u t n are much more intricate than those in the associative case

gradings on simple jordan and lie algebras abstract in this paper we describe all group gradings by a finite abelian group g of several types of simple jordan and lie algebras over an algebraically closed field f of characteristic zero

on z2 graded identities of ut2 e and their growth abstract let f be an infinite field of characteristic different from two and e be the infinite dimensional grassmann algebra over f we consider the upper triangular matrix algebra ut 2 e with entries in e endowed with the z 2 grading inherited by the natural z 2 grading of e and we study its ideal of z 2 graded polynomial identities t z 2 ideal and its relatively free algebra in particular we show that the set of z 2 graded polynomial identities of ut 2 e does not depend on the characteristic of the field moreover we compute the z 2 graded hilbert series of ut 2 e and its z 2 graded gelfand kirillov dimension

geometric aspects of fleming viot and dawson watanabe processes this paper is concerned with the intrinsic metrics of the two main classes of superprocesses for the fleming viot process we identify it as the bhattacharya distance and for dawson watanabe processes we find the kakutani hellinger metric the corresponding geometries are studied in some detail in particular representation formulas for geodesics and arc length functionals are obtained the relations between the two metrics yield a geometric interpretation of the identification of the fleming viot process as a dawson watanabe superprocess conditioned to have total mass 1 as an application a functional limit theorem for super brownian motion conditioned on local extinction is proved

a graph theoretic approach to graded identities for matrices abstract we consider the algebra m k c of k by k matrices over the complex numbers and view it as a crossed product with a group g of order k by imbedding g in the symmetric group s k via the regular representation and imbedding s k in m k c in the usual way this induces a natural g grading on m k c which we call a crossed product grading this grading is the so called elementary grading defined by any k tuple g 1 g 2 g k of distinct elements g i g we study the graded polynomial identities for m k c equipped with a crossed product grading to each multilinear monomial in the free graded algebra we associate a directed labeled graph this approach allows us to give new proofs of known results of bahturin and drensky on the generators of the t ideal of identities and the amitsur levitsky theorem our most substantial new result is the determination of the asymptotic formula for the g graded codimension of m k c

cheeger harmonic functions in metric measure spaces revisited let x d be a complete metric measure space with a locally doubling measure that supports a local weak l2 poincare inequality by assuming a heat semigroup type curvature condition we prove that cheeger harmonic functions are lipschitz continuous on x d gradient estimates for cheeger harmonic functions and solutions to a class of non linear poisson type equations are presented

mappings of bmo bounded distortion this paper can be viewed as a sequel to the work 9 where the theory of mappings of bmo bounded distortion is developed largely in even dimensions using singular integral operators and recent developments in the theory of higher integrability of jacobians in hardy orlicz spaces in this paper we continue this theme refining and extending some of our earlier work as well as obtaining results in new directions the planar case was studied earlier by g david 4 in particular he obtained existence theorems modulus of continuity estimates and bounds on area distortion for mappings of bmo distortion in fact in slightly more generality we obtain similar results in all even dimensions one of our main new results here is the extension of the classical theorem of painleve concerning removable singularties for bounded analytic functions to the class of mappings of bmo bounded distortion the setting of the plane is of particular interest and somewhat more can be said here because of the existence theorem or the measurable riemann mapping theorem which is not available in higher dimensions we give a construction to show our results are qualitatively optimal another surprising fact is that there are domains which support no bounded quasiregular mappings but admit

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

lipschitz algebras and derivations ii exterior differentiation abstract basic aspects of differential geometry can be extended to various non classical settings lipschitz manifolds rectifiable sets sub riemannian manifolds banach manifolds wiener space etc although the constructions differ in each of these cases one can define a module of measurable 1 forms and a first order exterior derivative we give a general construction which applies to any metric space equipped with a finite measure and produces the desired result in all of the above cases it agrees with cheeger s construction when the latter is defined it also applies to an important class of dirichlet spaces where however the known first order differential calculus in general differs from ours although the two are related

group gradings on full matrix rings we study g gradings of the matrix ring mn k k a field and give a complete description of the gradings where all the elements ei j are homogeneous called good gradings among these we determine the ones that are strong gradings or crossed products if g is a finite cyclic group and k contains a primitive g th root of 1 we show how all g gradings of mn k can be produced in particular we give a precise description of all c2 gradings of m2 k and show that for algebraically closed k any such grading is isomorphic to one of the two good gradings

gradings on the algebra of upper triangular matrices and their graded identities abstract let k be an infinite field and let ut n k denote the algebra of n n upper triangular matrices over k we describe all elementary gradings on this algebra further we describe the generators of the ideals of graded polynomial identities of ut n k and we produce linear bases of the corresponding relatively free graded algebras we prove that one can distinguish the elementary gradings by their graded identities we describe bases of the graded polynomial identities in several typical cases although in these cases we consider elementary gradings by cyclic groups the same methods serve for elementary gradings by any finite group

azumaya like properties of verbally prime algebras is a p i for a if and only if f or g is he proves over a field f of characteristic zero that a verbally prime algebra must be p i equivalent to one of the following 1 0 the trivial case or 2 m f ii x n matrices over the field f or 3 m e ii x ii matrices over an infinite dimensional grassmann algebra e or 4 m which we will not define or 5 the free algebra a case we shall ignore by always taking a to be pi since 3 a number of papers have appeared in which pi properties of n x n matrices have been generalized to verbally prime algebras one such paper is 4 in which razmyslov studies the trace identities of mk and the central polynomials of mk and m e the present paper grew out of an attempt to generalize the artin procesi theorem cf a to the verbally prime case we found that all verbally prime p i algebras satisfy weaker versions of the basic properties of azumaya algebras we will show as straightforward applications of kemer s and razmyslov s results

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

a differentiable structure for metric measure spaces abstract the main result of this paper is the provision of conditions under which a metric measure space admits a differentiable structure this differentiable structure gives rise to a finite dimensional l cotangent bundle over the given metric measure space and then to a sobolev space h 1 p over the given metric measure space the latter which is reflexive for p 1 this extends results of cheeger geom funct anal 9 1999 3 428 to a wider collection of metric measure spaces

pauli gradings on lie superalgebras and graded codimension growth abstract we introduce grading on certain finite dimensional simple lie superalgebras of type p t by elementary abelian 2 group this grading gives rise to pauli matrices and is a far generalization of z 2 z 2 grading on lie algebra of 2 2 traceless matrices we use this grading for studying numerical invariants of polynomial identities of lie superalgebras in particular we compute graded pi exponent corresponding to pauli grading

mosco convergence and wiener measures for conductive thin boundaries abstract the mosco convergence of energy functionals and the convergence of associated wiener measures are proved for a domain with highly conductive thin boundary we obtain those results for matrix valued conductivities and a family of speed measures measures of the underlying domain in particular this family includes the lebesgue measure and the one which makes the energy functional superposition the expectation of the displacement of the associated processes close to the boundary goes to due to the explosion of the conductivity at the limit

graded identities of block triangular matrices abstract let f be an infinite field and u t d 1 d n be the algebra of upper block triangular matrices over f in this paper we describe a basis for the g graded polynomial identities of u t d 1 d n with an elementary grading induced by an n tuple of elements of a group g such that the neutral component corresponds to the diagonal of u t d 1 d n in particular we prove that the monomial identities of such algebra follow from the ones of degree up to 2 n 1 our results generalize for infinite fields of arbitrary characteristic previous results in the literature which were obtained for fields of characteristic zero and for particular g gradings in the characteristic zero case we also generalize results for the algebra u t d 1 d n c with a tensor product grading where c is a color commutative algebra generating the variety of all color commutative algebras

polynomial identities for graded tensor products of algebras abstract let k be a field char k 0 we study the polynomial identities satisfied by z 2 graded tensor products of t prime algebras regev and seeman proved that in a series of cases such tensor products are pi equivalent to t prime algebras they conjectured that this is always the case we deal here with the remaining cases and thus confirm regev and seeman s conjecture for some small algebras we can remove the restriction on the characteristic of the base field and we show that the behaviour of the corresponding graded tensor products is quite similar to that for the usual ungraded tensor products finally we consider graded tensor products also called commutation factors and their identities we show that regev s a b theorem holds for graded tensor products whenever the gradings are by finite abelian groups furthermore we study the pi equivalence of graded tensor products of t prime algebras

graded polynomial identities of matrices we consider g graded polynomial identities of the p p matrix algebra mp k over a field k of characteristic 0 graded by an arbitrary group g we find relations between the g graded identities of the g graded algebra mp k and the g h graded identities of the tensor product of mp k and the h graded algebra mq k with a fine h grading we also find a basis of the g graded identities of mp k with an elementary grading such that the identity component coincides with the diagonal of mp k

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

on z2 graded polynomial identities of the grassmann algebra abstract let f be a field of characteristic zero and let e be the grassmann algebra of an infinite dimensional f vector space l in this paper we study the z 2 graded polynomial identities of e with respect to any fixed z 2 grading such that l is an homogeneous subspace we found explicit generators for the ideal t 2 e of graded polynomial identities of e and we determine its cocharacter and codimension sequences