renormalization theory in four dimensional scalar fields i we present a renormalization group approach to the renormalization theory of 44 using techniques that have been introduced and used in previous papers and that lead to very simple methods to bound the coefficients of the effective potential and of the schwinger functions the main aim of this paper is to show how one can in this way obtain then bounds

group identities on units of group algebras abstract let u be the group of units of the group algebra fg of a group g over a field f suppose that either f is infinite or g has an element of infinite order we characterize groups g so that u satisfies a group identity under the assumption that g modulo the torsion elements is nilpotent this gives a complete classification of such groups for torsion groups this problem has already been settled in recent years

the freiheitssatz for poisson algebras abstract we prove the freiheitssatz for poisson algebras in characteristic zero we also give a new proof of the tameness of automorphisms for two generated free poisson algebras and show that an analogue of the commutator test theorem is equivalent to the two dimensional classical jacobian conjecture

graph rules for the linked cluster expansion of the legendre effective action graph rules for the linked cluster expansion of the legendre effective action are derived and proven in d 2 euclidean dimensions a key aspect is the weight assigned to articulation vertices which is itself shown to be computable from labeled tree graphs the hopping interaction is allowed to be long ranged and scale dependent thereby producing an in principle exact solution of s functional renormalization group equation graph rules for the linked cluster expansion of the legendre effective action are derived and proven in d 2 euclidean dimensions a key aspect is the weight assigned to articulation vertices which is itself shown to be computable from labeled tree graphs the hopping interaction is allowed to be long ranged and scale dependent thereby producing an in principle exact solution of s functional renormalization group equation

the repulsive lattice gas the independent set polynomial and the lovasz local lemma we elucidate the close connection between the repulsive lattice gas in equi librium statistical mechanics and the lovasz local lemma in probabilistic com binatorics we show that the conclusion of the lovasz local lemma holds for dependency graph g and probabilities px if and only if the independent set polynomial for g is nonvanishing in the polydisc of radii px furthermore we show that the usual proof of the lovasz local lemma which provides a suffi cient condition for this to occur corresponds to a simple inductive argument for the nonvanishing of the independent set polynomial in a polydisc which was discovered implicitly by shearer 98 and explicitly by dobrushin 37 38 we also present some refinements and extensions of both arguments including a generalization of the lovasz local lemma that allows for soft dependencies in addition we prove some general properties of the partition function of a repulsive lattice gas most of which are consequences of the alternating sign property for the mayer coefficients we conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs

enveloping algebras satisfying a polynomial identity in this paper we study ordinary and restricted enveloping algebras satisfying a polynomial identity we first show how the methods of j bergen and d s passman j algebra in press can be used to handle u l in all characteristics and u l if the latter ring is prime in particular we offer a simpler proof of the result of yu a bachturin j austral math soc 18 1974 10 21 on ordinary enveloping algebras in characteristic p 0 our main theorem asserts that in general u l is p i if and only if l has a restricted subalgebra of finite codimension which is essentially commutative these results are clearly the lie algebra analogs of the p i group ring theorems of m k smith j algebra 18 1971 477 499 and d s passman pacific j math 36 1971 467 483

convergence of mayer expansions the tree graph bound of battle and federbush is extended and used to provide a simple criterion for the convergence of iterated mayer expansions as an application estimates on the radius of convergence of the mayer expansion for the two dimensional yukawa gas nonstable interaction are obtained

character degrees of p groups and pro p groups in the 1970s isaacs conjectured that there should be a logarithmic bound for the length of solvability of a p group g with respect to the number of different irreducible character degrees of g so far there are just a few partial results for this conjecture in this note we say that a pro p group g has property i if there is a real number d d g that just depends on g such that for any open normal subgroup n d l g n log2 cd g n d we prove that any p adic analytic pro p group has property i we also study the first congruence subgroup g of a classical chevalley group g with respect to the local ring fpt we show that if lie g fp has a non degenerated killing form then g has property i

iterated mayer expansion for classical gases at low temperatures we derive iterated mayer expansions for classical gases and establish recursive bounds which control their convergence these bounds are useful for gases with two body forces which are strong and possibly attractive at distances that are short compared to their range our procedure is based on splitting the potential into pieces of decreasing strength and increasing range this may be called a renormalization group treatment of a classical gas we apply our results to yukawa lattice gas models and obtain convergence of series expansions for the pressure for a range of parameters temperature fugacities range of the interaction that was inaccessible before application to 3 dimensionalu 1 lattice gauge theory coulomb gas z ferromagnet will be made elsewhere

application of generalized thermostatistics to fully developed turbulence abstract we apply the formalism of non extensive statistical mechanics to fully developed turbulent flows we obtain analytical formulas for probability density functions of velocity differences depending on distance r and reynolds number r which are in very good agreement with turbulence experiments the skewness is taken into account by analysing universal contributions due to higher order correlations of the chaotic dynamics we also calculate the moment scaling exponents m which also agree well with experimental data

constructive renormalization for φ24 theory with loop vertex expansion in this paper we construct the two dimensional euclidean 4 quantum field theory using the method of loop vertex expansion we reproduce the results of standard constructive theory for example the borel summability of the schwinger functions in the coupling constant our method should be also suitable for the future construction of grosse wulkenhaar models on non commutative space time

a 5 engel associative algebra whose group of units is not 5 engel abstract let r be an associative ring with unity and let r and u r denote the associated lie ring with a b a b b a and the group of units of r respectively in 1983 gupta and levin proved that if r is a nilpotent lie ring of class c then u r is a nilpotent group of class at most c the aim of the present note is to show that in general a similar statement does not hold if r is n engel we construct an algebra r over a field of characteristic 2 3 such that the lie algebra r is 5 engel but the group u r is not

on lie solvable restricted enveloping algebras abstract in this note we study the lie derived lengths of a restricted enveloping algebra u l for a non abelian restricted lie algebra l over a field of positive characteristic p for p 2 we show that if the lie derived length of u l is minimal then u l is lie nilpotent moreover we investigate the case when the strong lie derived length of u l is minimal for odd p we establish a classification of lie centrally metabelian restricted enveloping algebras

mayer expansions and the hamilton jacobi equation we review the derivation of wilson s differential equation in infinitely many variables which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal integration out is performed we show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic the resulting series solutions are generalizations of the mayer expansion in statistical mechanics the differential equation approach gives a remarkable identity for connected parts and precise estimates which include criteria for convergence of iterated mayer expansions applications include the yukawa gas in two dimensions past the 4 threshold and another derivation of some earlier results of gopfert and mack

connected graded hopf algebras we study algebraic and homological properties of two classes of infinite dimensional hopf algebras over an algebraically closed field k of characteristic zero the first class consists of those hopf k algebras that are connected graded as algebras and the second class are those hopf k algebras that are connected as coalgebras for many but not all of the results presented here the hopf algebras are assumed to have finite gel fand kirillov dimension it is shown that if the hopf algebra h is a connected graded hopf algebra of finite gel fand kirillov dimension n then h is a noetherian domain which is cohen macaulay artin schelter regular and auslander regular of global dimension n it has s2 idh and is calabi yau detailed information is also provided about the hilbert series of h our results leave open the possibility that the first class of algebras is properly contained in the second for this second class the hopf k algebras of finite gel fand kirillov dimension n with connected coalgebra the underlying coalgebra is shown to be artin schelter regular of global dimension n both these classes of hopf algebra share many features in common with enveloping algebras of finite dimensional lie algebras for example an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra nevertheless we construct as one of our main results an example of a hopf k algebra h of gel fand kirillov dimension 5 which is connected graded as an algebra and connected as a coalgebra but is not isomorphic as an algebra to u g for any lie algebra g

nonmatrix varieties and nil generated algebras whose units satisfy a group identity abstract letr denote the group of units of an associative algebrarover an infinite fieldf we prove that ifris unitarily generated by its nilpotent elements thenr satisfies a group identity precisely whenrsatisfies a nonmatrix polynomial identity as an application we examine the group algebrafgof a torsion groupgand the restricted enveloping algebrau l of ap nil restricted lie algebral giambruno sehgal and valenti recently proved that if the group of units fg satisfies a group identity thenfgsatisfies a polynomial identity thus confirming a conjecture of brian hartley we show that in fact fg satisfies a group identity if and only iffgsatisfies a nonmatrix polynomial identity in the case of restricted enveloping algebras we prove thatu l satisfies a group identity if and only ifu l satisfies the engel condition

a pi degree theorem for quantum deformations abstract we prove that if a filtered quantization a of a finitely generated commutative domain over a field k is a pi algebra then a is commutative if char k 0 and its pi degree is a power of p if char k p

some properties on rings with units satisfying a group identity abstract for convenience a ring with units satisfying a group identity will be called a gi ring we show that gi rings have the following properties which are also properites of pi rings 1 any gi ring is dedekind finite von neumann finite 2 nilpotent elements of a semiprimitive gi ring have bounded index 3 the kurosh problem has a positive answer for gi algebras namely any algebraic gi algebra is locally finite we also study hartley s problem for algebraic gi algebras

grad o perturbations of massless gaussian fields we investigate weak perturbations of the continuum massless gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

characterization of p adic analytic groups in terms of wreath products abstract we show that a finitely generated pro p group is p adic analytic i e can be given the structure of a lie group over q p if and only if it does not involve arbitrarily large wreath products of the form c p wr c p n this result whose proof applies zelmanov s recent solution to the restricted burnside problem is in fact equivalent to zelmanov s theorem

prime ideals in differential operator rings and crossed products of infinite groups abstract we use a version of the martindale ring of quotients to study prime ideals in extensions of a ringr corresponding to two cases ifg is a group of automorphisms ofr we form the crossed product r g if g is a lie algebra of derivations of r we have the twisted differential operator ring denoted byr g sometimes known as the twisted smash product r t u g we obtain analogues of incomparability for crossed products of nilpotent groups and differential operator rings of solvable lie algebras in the case of crossed products the incomparability result of d s passman and m lorenz for the infinite cyclic group also proved by g bergman is generalized

symanzik s improved actions from the viewpoint of the renormalization group we investigate symanzik s improvement program in a four dimensional euclidean scalar field theory with smooth momentum space cutoff we use wilson s renormalization group transformation to define the improved actions as a sequence of initial data for the effective action at the fundamental cutoff this leads to a sequence of solutions to the renormalization group equation we define the parameters of the improved actions implicitly by conditions on the effective action at a renormalization scale the improved actions are close approximations to the continuum effective action we prove their existence to every order of improvement and to every order of renormalized perturbation theory

a connected graph identity and convergence of cluster expansions recently fernandez and procacci found an estimate on the radius of convergence of the cluster expansion for a gas of particles on a discrete space their estimate is an improvement on previous results of koteck and preiss and of dobrushin this note presents a connected graph identity and uses it to prove a version of their estimate that applies to considerably more general contexts including that where the space is not discrete

delta methods in enveloping rings abstract let l be a k lie algebra possibly restricted and define then l is a restricted lie ideal of l which is clearly the lie algebra analog of the finite conjugate center of a group indeed in analogy with known results on group algebras and crossed products we show that l controls certain properties of enveloping rings determined by l specifically we consider the annihilator ideals and primeness of various twisted smash products r u l or r u l furthermore we study x inner automorphisms x inner derivations intersection theorems and primitivity of the twisted enveloping algebras u l or u l

an iterated mayer expansion for the yukawa gas we construct a convergent expansion for the yukawa gas at small activity and inverse temperature 4 e 2

identities on units of algebraic algebras let a be an algebraic algebra over an infinite field k and let u a be its group of units we prove a stronger version of hartley s conjecture for a namely if a laurent polynomial identity lpi for short f 0 is satisfied in u a then a satisfies a polynomial identity pi we also show that if a is non commutative then a is a pi ring provided f 0 is satisfied by the non central units of a in particular a is locally finite and thus the kurosh problem has a positive answer for k algebras whose unit group is lpi moreover f 0 holds in u a if and only if the same identity is satisfied in a the last fact remains true for generalized laurent polynomial identities provided that a is locally finite

group identities on the units of algebraic algebras with applications to restricted enveloping algebras abstract an algebra a is called a gi algebra if its group of units a satisfies a group identity we provide positive support for the following two open problems 1 does every algebraic gi algebra satisfy a polynomial identity 2 is every algebraically generated gi algebra locally finite

polynomial identities in graded group rings restricted lie algebras and p adic analytic groups let g be any finitely generated group and let k be a field of characteristic p 0 it is shown that the graded group ring gr kg satisfies a nontrivial polynomial identity if and only if the pro p completion of g is p adic analytic i e can be given the structure of a lie group over the p adic field q p the proof applies theorems of lazard quillen and passman as well as results on engel lie algebras and on dimension subgroups in positive characteristic

poisson pi algebras we study poisson algebras satisfying polynomial identities in particular such algebras satisfy customary identities farkas 1998 1999 our main result is that the growth of the corresponding codimensions of a poisson algebra with a nontrivial identity is exponential with an integer exponent we apply this result to prove that the tensor product of poisson pi algebras is a pi algebra we also determine the growth of the poisson grassmann algebra and of the hamiltonian algebras

polynomial and poisson dependence in free poisson algebras and free poisson fields abstract we prove that any two poisson dependent elements in a free poisson algebra and a free poisson field of characteristic zero are algebraically dependent thus answering positively a question from makar limanov and umirbaev 2007 8 we apply this result to give a new proof of the tameness of automorphisms for free poisson algebras of rank two see makar limanov and umirbaev 2011 9 makar limanov et al 2009 10

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

the nilpotency of the radical in a finitely generated p i ring abstract let r x 1 x k be a p i ring satisfying a monic polynomial identity one of its coefficients is 1 where is a central noetherian subring it is proven that n r the nil radical of r is nilpotent as a corollary by taking f a field we settle affirmatively the open problem posed in c procesi rings with polynomial identities p 186 marcel dekker new york 1973 we prove the jacobson radical of a finitely generated p i algebra is nilpotent

dimension subgroups nilpotency indices and the number of generators of ideals in p group algebras abstract various problems in modular p group algebras are solved through extensive study of dimension subgroups as an application it is shown that if g is an infinite res p group p 2 and k is a field of characteristic p then the least upper bound on the numbers of generators of ideals in kg equals the minimal index of a cyclic subgroup of g

massless sine gordon and massive thirring models proof of coleman s equivalence we prove coleman s conjecture on the equivalence between the massless sine gordon model with finite volume interaction and the thirring model with a finite volume mass term

a group theoretic characterization of linear groups let r be a group when is f linear this is an old problem the first to study this question systematically was malcev in 1940 m who essentially reduced the problem to finitely generated groups note that a finitely generated group is linear over some field of characteristic zero if and only if it can be embedded in cl c for some n very little progress was made since that paper of malcev although as linear groups are a quite special type of group many necessary conditions were obtained e g r should be residually finite and even virtually residually p for almost all primes p f should be virtually torsion free and if not solvable by finite it has a free non abelian subgroup etc cf z of course none of these properties characterizes the finitely generated linear groups over in this paper we give such a characterization using the congruence structure of r first some definitions for a group h d h denotes the minimal number of generators for h

on group identities for the unit group of algebras and semigroup algebras over an infinite field let k be an infinite field we fully describe when the unit group of a semigroup algebra k s of a semigroup s generated by finitely many periodic elements satisfies a group identity this and some other recent results are proved by first showing that semiprime k algebras generated by units are necessarily reduced whenever their unit group satisfies a group identity

perfect and semiperfect restricted enveloping algebras abstract for a restricted lie algebra l the conditions under which its restricted enveloping algebra u l is semiperfect are investigated moreover it is proved that u l is left or right perfect if and only if l is finite dimensional

computing the symmetric ring of quotients abstract we discuss and compute the symmetric martindale ring of quotients for various classes of prime rings in particular we consider free algebras and group algebras

extensions of valuation rings in central simple algebras certain subrings r of simple algebras q finite dimensional over their center k are studied these rings are called q valuation rings since they share many properties with commutative valuation rings let v be a valuation ring of k the center of q and let m be the set of q valuation rings r in q with r n k v then 1s 1 this extension theorem which does not hold if one considers only total valuation rings was proved by n i dubrovin here first a somewhat different proof of this result is given and then information about the set m is obtained theorem the elements in j are conjugate if v has finite rank theorem the elements in m are total valuation rings if m contains one total valuation ring in this case q is a division ring theorem i i 1 if 5 contains an invariant total valuation ring

a lie algebra generalization of the amitsur levitski theorem 0 1 if r is any algebra over a field f and xi e r i l k let iix 1 y 1 e r denote the alternating sum of products z sg 0 x x over all permutations 0 of l k one says that r satisfies the k fold standard identity if ix xk 0 for any choice xi e r of course if pc r is any subspace of r one can restrict one s attention to p and say that p satisfies the k fold standard identity if 1 x1 x 0 for any choice xi f p now the amitsur levitski theorem asserts that the n x n matrix algebra m n f satisfies the 2n fold standard identity for a survey and history of this theorem see 9 1 see also 17 i it is of course enough to consider the case where f c some time ago see 3 we came upon a proof of the amitsur levitski theorem from the point of lie algebra cohomology the cohomology of m n c as a lie algebra is an exterior algebra with primitive generators having degrees 1 2 2n 1 the failure of m n cc to have a primitive class in degree 2n 1 leads to the trace relation tr x x 0 for any xi e m n c for i i 2n 1 but this quickly implies the amitsur levitski theorem in fact the argument can be pushed further if d g m n cc denotes the lie algebra of all n x n skew symmetric matrices one knows that d has no primitive class in degree 2n 1 in case n is even this led to a new identity in 3 1 for iz even

on units of twisted group algebras abstract we study units of twisted group algebras let g be a finite group and k be a field of characteristic p 0 assume that k is not algebraic over a finite field we determine when units of k t g do not contain any nonabelian free subgroup we also discuss what will happen when g is locally finite for twisted group algebras of locally finite groups over any infinite field of characteristic p 0 we characterize those twisted group algebras with units satisfying a group identity finally we include a new characterization for twisted group algebras to satisfy a polynomial identity

matrix rings and linear groups over a field of fractions of enveloping algebras and group rings i 1 1 let h be a lie algebra over a field k u h be its universal envelope it is well known that u h is a domain see 1 section v 31 and the theorem of cohn see 2 states that u h can be imbedded in a field when h is locally finite u h has even a field of fractions see 1 theorem v 3 61 the class of lie algebras whose universal envelope has a field of fractions is however much wider than the class of locally finite lie algebras it follows from proposition 4 1 of the article that it contains for instance all the locally soluble by locally finite algebras this gives a negative answer of the question in 2 page 528 let h satisfy any one of the following three conditions

infinite crossed products and group graded rings in this paper we precisely determine when a crossed product r g is semiprime or prime indeed we show that these conditions ultimately depend upon the analogous conditions for the crossed products r n of the finite subgroups n of g and upon the interrelationship between the normalizers of these subgroups and the ideal structure of r the proof offered here is combinatorial in nature using the a methods and is entirely self contained furthermore since the argument applies equally well to strongly g graded rings we have opted to work in this more general context let g be a multiplicative group and let r be a ring with 1 then a crossed product r g of g over r is an associative ring determined by g r and certain other parameters to be more precise for each x e g there exists an element je r g and every element a e r g is uniquely writable as a finite sum

nontrivial fixed points and screening in the hierarchical two dimensional coulomb gas we show the existence and asymptotic stability of two fixed points of the renormalization group transformation for the hierarchical two dimensional coulomb gas in the sine gordon representation and temperatures slightly greater than the critical one we prove also that the correlations at the fixed points decay as in the hierarchical massive scalar free theory that is asdxy 4 we argue that this is the natural definition of screening in the hierarchical approximation

the two dimensional o n nonlinear σ model renormalisation and effective actions we establish the existence of the wilson renormalised trajectory of theo n sigma model in perturbation theory in the effective charge this yields a proof of perturbative renormalisability and is also relevant in the small field analysis of the rigorous renormalisation group construction of the continuum theory

corrected loop vertex expansion for φ24 theory this paper is an extended erratum to 1 in which the classic construction and borel summability of the 4 euclidean quantum eld theory was revisited combining a multi scale analysis with the constructive method called loop vertex expansion lve unfortunately we discovered an important error in the method of 1 we explain the mistake and provide a new correct construction of the 4 theory according to the lve

renormalization group flow of a hierarchical sine gordon model by partial differential equations we use a renormalization group differential equation to rigorously control the renormalization group flow in a hierarchical lattice sine gordon field theory in the kosterlitz thouless phase

the lie structure of enveloping algebras abstract this paper examines the lie structure of restricted universal enveloping algebras u l over fields of characteristic p 0 it is determined precisely when u l considered as a lie algebra is soluble for p 2 nilpotent or satisfies the engel condition

on groups of polynomial subgroup growth let be a finitely generated group andan the number of its subgroups of indexn we prove that assuming is residually nilpotent e g linear thenan grows polynomially if and only if is solvable of finite rank this answers a question of segal the proof uses a new characterization ofp adic analytic groups the theory of algebraic groups and the prime number theorem the method can be applied also to groups of polynomial word growth

delta methods in enveloping algebras of lie superalgebras let l be a lie superalgebra over a field k of characteristic 2 we define a l el dima l oo then a l is a lie ideal of l and is restricted if l is restricted a l is the lie superalgebra analog of the lie delta ideal used by the authors in the study of enveloping rings and also of the finite conjugate center of a group used in the study of group algebras and crossed products in this paper we examine u l where depending upon char k u l denotes either the enveloping algebra or the restricted enveloping algebra of l we show that a l controls certain properties of u l specifically we consider semiprimeness primeness almost constants almost centralizers central closures and the artinian condition

lie structure of truncated symmetric poisson algebras abstract the paper naturally continues series of works on identical relations of group rings enveloping algebras and other related algebraic structures let l be a lie algebra over a field of characteristic p 0 consider its symmetric algebra s l n 0 u n u n 1 which is isomorphic to a polynomial ring it also has a structure of a poisson algebra where the lie product is traditionally denoted by this bracket naturally induces the structure of a poisson algebra on the ring s l s l x p x l which we call a truncated symmetric poisson algebra we study lie identical relations of s l namely we determine necessary and sufficient conditions for l under which s l is lie nilpotent strongly lie nilpotent solvable and strongly solvable where we assume that p 2 to specify the solvability we compute the strong lie nilpotency class of s l also we prove that the lie nilpotency class coincides with the strong lie nilpotency class in case p 3 shestakov proved that the symmetric algebra s l of an arbitrary lie algebra l satisfies the identity x y z 0 if and only if l is abelian we extend this result for the strong lie nilpotency and the strong solvability of s l we show that the solvability of s l and s l in case char k 2 is different from other characteristics namely we construct examples of such algebras which are solvable but not strongly solvable we use delta sets for lie algebras and the theory of identical relations of poisson algebras also we study filtrations in poisson algebras and prove results on products of terms of the lower central series for poisson algebras

lie solvable enveloping algebras of characteristic two abstract lie solvable restricted enveloping algebras were characterized by riley and shalev except when the ground field is of characteristic 2 we resolve the characteristic 2 case here which completes the classification as an application of our result we obtain a characterization of ordinary lie algebras over any field whose enveloping algebra is lie solvable

the tame and the wild automorphisms of polynomial rings in three variables let c f x1 x2 xn be the polynomial ring in the variables x1 x2 xn over a field f and let autc be the group of automorphisms of c as an algebra over f an automorphism autc is called elementary if it has a form x1 xi 1 xi xi 1 xn 7 x1 xi 1 xi f xi 1 xn where 0 6 f f f x1 xi 1 xi 1 xn the subgroup of autc generated by all the elementary automorphisms is called the tame subgroup and the elements from this subgroup are called tame automorphisms of c non tame automorphisms of the algebra c are called wild it is well known 6 9 10 11 that the automorphisms of polynomial rings and free associative algebras in two variables are tame at present a few new proofs of these results have been found see 5 8 however in the case of three or more variables the similar question was open and known as the generation gap problem 2 3 or tame generators problem 8 the general belief was that the answer is negative and there were several candidate counterexamples see 5 8 12 7 19 the best known of them is the following automorphism aut f x y z constructed by nagata in 1972 see 12 x x x yz z y y 2 x yz x x yz z z z

log sobolev inequality for the continuum sine gordon model we prove a multiscale generalisation of the bakry emery criterion for a measure to satisfy a log sobolev inequality our criterion relies on the control of an associated pde well known in renormalisation theory the polchinski equation it implies the usual bakry emery criterion but we show that it remains effective for measures which are far from log concave indeed using our criterion we prove that the massive continuum sine gordon model with beta 6 pi satisfies asymptotically optimal log sobolev inequalities for glauber and kawasaki dynamics these dynamics can be seen as singular spdes recently constructed via regularity structures but our results are independent of this theory

lie identities on enveloping algebras of restricted lie superalgebras abstract let l be a restricted lie superalgebra with its restricted enveloping algebra u l over a field f of characteristic p 2 then u l can be viewed both as a lie algebra and a lie superalgebra we characterize l when u l is lie solvable lie nilpotent or lie super nilpotent

loop vertex expansion for φ2k theory in zero dimension in this paper we extend the method of loop vertex expansion to interactions with degree higher than 4 as an example we provide through this expansion an explicit proof that the free energy of 2k scalar theory in zero dimension is borel le roy summable of order k 1 we detail the computations in the case of a 6 interaction in this paper we extend the method of loop vertex expansion to interactions with degree higher than 4 as an example we provide through this expansion an explicit proof that the free energy of 2k scalar theory in zero dimension is borel le roy summable of order k 1 we detail the computations in the case of a 6 interaction

generalized constructive tree weights the loop vertex expansion lve is a quantum field theory qft method which explicitly computes the borel sum of feynman perturbation series this lve relies in a crucial way on symmetric tree weights which define a measure on the set of spanning trees of any connected graph in this paper we generalize this method by defining new tree weights they depend on the choice of a partition of a set of vertices of the graph and when the partition is non trivial they are no longer symmetric under permutation of vertices nevertheless we prove they have the required positivity property to lead to a convergent lve in fact we formulate this positivity property precisely for the first time our generalized tree weights are inspired by the brydges battle federbush work on cluster expansions and could be particularly suited to the computation of connected functions in qft several concrete examples are explicitly given

when is a power of the frobenius map on a noncommutative ring a homomorphism abstract let x x p be the frobenius map on an associative unital ring r with prime characteristic p it is well known that whenever r is commutative n is a ring homomorphism for all positive integers n the converse however is not true in general indeed we prove that if r is m engel for some positive integer m then there exists a positive integer n 0 depending only on m such that for all n n 0 n is a ring homomorphism with central image conversely if any one of the following conditions holds n respects addition n respects multiplication n respects lie multiplication or the image of n is commutative then r is m engel for some m depending only on n consequently if is surjective and any one of the aforementioned conditions holds then r must be commutative

quantum evolution of power law mixed states we study mixed quantum states described by a statistical operator bn n real with b quadratic in the position and momentum operators these states are parameterized as density matrices exhibiting the maximum q entropy q maxent form they can be regarded as the mixed state counterpart of the power law tail wave packets recently introduced by lillio and mantegna phys rev lett 84 2000 1061 84 2000 4516 we study the time evolution of these q maxent density matrices we also investigate the main features of their associated position probability distributions in particular we focus attention on the convergence of the concomitant moments

compact groups in which all elements have countable right engel sinks a right engel sink of an element g of a group g is a set mathscr r g such that for every x in g all sufficiently long commutators g x x dots x belong to mathscr r g thus g is a right engel element precisely when we can choose mathscr r g 1 it is proved that if every element of a compact hausdorff group g has a countable right engel sink then g has a finite normal subgroup n such that g n is locally nilpotent

lower central series of free algebras in symmetric tensor categories abstract we continue the study of the lower central series of a free associative algebra initiated by feigin and shoikhet 2007 fs we generalize via schur functors the constructions of the lower central series to any symmetric tensor category specifically we compute the modified first quotient b 1 and second and third quotients b 2 and b 3 of the series for a free algebra t v in any symmetric tensor category generalizing the main results of feigin and shoikhet 2007 fs and arbesfeld and jordan 2010 aj in the case a m n t c m n we use these results to compute the explicit hilbert series finally we prove a result relating the lower central series to the corresponding filtration by two sided associative ideals confirming a conjecture from etingof et al 2009 ekm and another one from arbesfeld and jordan 2010 aj as corollaries