non linearizable algebraic k actions on affine spaces non linearizable faithful algebraic r r actions of torus on the real affine n space r n are given for each r 1 n 3 when n 4

sum rules and perfect screening conditions for the one component plasma first we show from the bbgky hierarchy and under exponential clustering assumption that the three dimensional one component plasma obeys the compressibility sum rule which links the compressibility to the fourth moment of the two point correlation function then it is proved that the first equation of the hierarchy is equivalent to the known value of the correlation function of momentum current and density finally we are concerned with the energy density its definition and its correlation with the numerical density and with itself

subgroups of polynomial automorphisms with diagonalizable fibers let r be an integral domain over a field k and g a subgroup of the automorphism group of the polynomial ring r x1 xn over r in this paper we discuss when g is diagonalizable under the assumption that g is diagonalizable over the field of fractions of r we are particularly interested in the case where g is a finite abelian group kraft and russell 2014 8 imply that every finite abelian subgroup of autrr x1 x2 is diagonalizable if r is an affine pid over k c one of the main results of this paper says that the same holds for a pid r over any field k containing enough roots of unity

stable tameness of two dimensional polynomial automorphisms over a regular ring abstract in this paper it is established that all two dimensional polynomial automorphisms over a regular ring r are stably tame this results from the main theorem of this paper which asserts that an automorphism in any dimension n is stably tame if said condition holds point wise over spec r a key element in the proof is a theorem which yields the following corollary over an artinian ring a all two dimensional polynomial automorphisms having jacobian determinant one are stably tame and are tame if a is a q algebra another crucial ingredient of interest in itself is that stable tameness is a local property if an automorphism is locally tame then it is stably tame

sum rules for correlation functions of ionic mixtures in arbitrary dimension d 2 the correlations in classical multi component ionic mixtures with a spatial dimension d 2 are studied by using a restricted grand canonical ensemble and the associated hierarchy equations for the correlation functions sum rules for the first few moments of the two particle correlation function are derived and their dependence on d is established by varying d continuously near d 2 it is shown how the sum rules for the two dimensional mixture are related to those for mixtures at higher d

simple birational extensions of the polynomial algebra c 3 the abhyankar sathaye problem asks whether any biregular embedding c k c n can be rectified that is whether there exists an automorphism a e aut c n such that a o is a linear embedding here we study this problem for the embeddings c 3 c 4 whose image x c 3 is given in c 4 by an equation p f x y u g x y z 0 where f c x y 0 and g e c x y z under certain additional assumptions we show that indeed the polynomial p is a variable of the polynomial ring c 4 c x y z u i e a coordinate of a polynomial automorphism of c 4 this is an analog of a theorem due to sathaye 1976 which concerns the case of embeddings c 2 c 3 besides we generalize a theorem of miyanishi 1984 giving for a polynomial p as above a criterion for when x p 1 0 c 3

structure of a fibrations over one dimensional seminormal semilocal domains abstract in this paper we give a structure theorem for an a fibration over a one dimensional noetherian seminormal semilocal domain and show that in this situation any a fibration whose spectrum occurs as an affine open subscheme of the spectrum of an a 1 fibration equivalently an affine line a 1 is actually a the structure theorem provides examples of a fibrations over one dimensional noetherian seminormal semilocal domains whose spectra are not affine open subschemes of any affine line a 1 over the base ring we also construct examples of nontrivial a fibrations over one dimensional noetherian non seminormal local domains whose spectra are open subschemes of a 1 fibrations over the base ring

derivations of r x y z with a slice the question under consideration is whether every locally nilpotent r derivation of r x y z with a slice has kernel a generated by two elements over r where r is a polynomial ring over a field of characteristic zero theorem 1 1 gives a fundamental property of such kernels namely that a is an a2 fibration over r while it is an open question whether every a2 fibration over r is trivial the property of a asserted in the theorem is necessary to the condition that a is a polynomial ring in two variables over r the last section of the paper presents a family of examples n n 1 which are quite simple to define but whose status relative to the kernel question is not known

classical nonlinear response of a chaotic system i collective resonances we develop a general semiquantitative picture of nonlinear classical response in strongly chaotic systems in contrast to behavior in integrable or almost integrable systems the nonlinear classical response in chaotic systems vanishes at long times the exponential decay of the response functions in the case of strong chaos is attributed to both exponentially decaying and growing elements in the stability matrices we calculate the linear and second order response in one of the simplest chaotic systems free classical motion on a compact surface of constant negative curvature the response reveals certain features of collective resonances which do not correspond to any periodic classical trajectories we demonstrate the relevance of the model for the interpretation of spectroscopic experiments

homogeneous locally nilpotent derivations having slices and embeddings of affine spaces abstract let m and n be positive integers such that n m and let b be a polynomial ring in m n 1 variables over a field k of characteristic 0 we give a bijective correspondence between the equivalence classes of embeddings a m a n and the equivalence classes of sequences of mutually commuting locally nilpotent derivations i 1 i m on b in some form which are homogeneous with respect to a z grading on b and have slices the intersection a of the kernels of i for 1 i m inherits the z grading on b we show that a is a polynomial ring with homogeneous coordinates if and only if the corresponding embedding is rectifiable

sum rules and perfect screening conditions for the 2d one component plasma first we show from the bbgky hierarchy and under exponential clustering assumption that the three dimensional one component plasma obeys the compressibility sum rule which links the compressibility to the fourth moment of the two point correlation function then it is proved that the first equation of the hierarchy is equivalent to the known value of the correlation function of momentum current and density finally we are concerned with the energy density its definition and its correlation with the numerical density and with itself

discriminants and automorphism groups of veronese subrings of skew polynomial rings we study important invariants and properties of the veronese subalgebras of q skew polynomial rings including their discriminant center and automorphism group as well as cancellation property and the tits alternative

an algorithmic proof of suslin s stability theorem for polynomial rings abstract let k be a field then gaussian elimination over k and the euclidean division algorithm for the univariate polynomial ring k x allow us to write any matrix in sln k or sln k x n 2 as a product of elementary matrices suslin s stability theorem states that the same is true for sln k xl xm with n 3 and m 1 in this paper we present an algorithmic proof of suslin s stability theorem thus providing a method for finding an explicit factorization of a given polynomial matrix into elementary matrices grobner basis techniques may be used in the implementation of the algorithm

classical chaos and fluctuation dissipation relations for nonlinear response the classical nonlinear optical response is expressed in a form that closely resembles the fluctuation dissipation theorem the nth order response is shown to depend on interferences among n closely lying trajectories the relevant dynamical information on the vicinity of a given trajectory can be recast using the stability matrix related to the lyapunov exponents no such interference exists in the linear response and the nonlinear response is consequently a much more sensitive probe for classical chaos sequences of multiple femtosecond pulses can be designed to directly probe the stability matrix textcopyright 1996 the american physical society

fixed point free locally nilpotent derivations of a2 fibrations abstract let r be a regular ring containing q and let a be an a 2 fibration over r we prove in this paper that every fixed point free locally nilpotent r derivation of a has a slice as a consequence an a 2 fibration over a factorial regular ring r is trivial if and only if it has a fixed point free locally nilpotent r derivation our results answer a question raised by freudenburg 2009 11 question 2 which actually was the main motivation of this work

equilibrium properties of a multi component ionic mixture i sum rules for correlation functions abstract equilibrium statistical methods are used to derive sum rules for two and three particle correlation functions of a multi component ionic mixture some of these rules are general consequences of the electrostatic character of the interaction whereas others depend on specific thermodynamic properties of the system the first group of rules follows from the bbgky hierarchy and a clustering hypothesis for ursell functions the sum rules of the second group are obtained by describing the system with the help of a restricted grand canonical ensemble in which the particle numbers of the various components in the mixture fluctuate under the condition that the total charge in the system remains constant

multidimensional measures of impulsively driven stochastic systems based on the kullback leibler distance by subjecting a dynamical system to a series of short pulses and varying several time delays we can obtain multidimensional characteristic measures of the system multidimensional kullback leibler response function klrf which are based on the kullback leibler distance between the initial and final states are defined we compare the klrf which are nonlinear in the probability density with ordinary response functions obtained from the expectation value of a dynamical variable which are linear we show that the klrf encode different level of information regarding the system s dynamics for overdamped stochastic dynamics two dimensional klrf shows a qualitatively different variation with the time delays between pulses depending on whether the system is initially in a steady state or in thermal equilibrium

serre s problem for one dimensional ground rings in 131 ming chang kang considered the following question l et r be a commutative noetherian one dimensional ring connected and with finite normalization and let a r x x be the polynomial ring is every bnitely generated projective a module isomorphic to a direct sum of a rank one module and a free module the condition that r be connected is not essential once the result is proved for the connected case it is easily generalized to the case where r is a product of several connected rings if r is a pid this indeed is serre s conjecture and also for r regular i e dedekind the result follows from quillen s localization theorem 4 and the decomposition properties of projective modules over dedekind rings kang solved the question in the affirmative in almost every case the only exception being the case 4 g r and the rank of the projective a module equal to 2 his results extended former partial results of seshadri and murthy see references in 3 for a r x the result follows from plumstead s solution of the first eisenbud evans conjecture 8 here we will till in the gap and prove that the question above always has the answer yes by means of an extension and simplification of kang s arguments which systematically exploits the theory of seminormal rings due to traverso swan and others by standard arguments one may suppose r reduced and connected the case of seminormal r will be handled by means of suslin s results quoting s to perform a reduction to this latter case we seminormalize r and use glueing milnor s theorem to prove the result some questions and remarks conclude the note

on zariski s cancellation problem in positive characteristic abstract in this paper we show that when k is a field of positive characteristic the affine space a k n is not cancellative for any n 3

an invariant structure of the multi particle correlations of the two dimensional one component plasma the model under consideration is that of a two dimensional one component plasma confined to the surface of a sphere in the canonical format the system is mapped onto a discrete one dimensional anticommuting field theory a unitary transformation of anticommuting variables is revealed to generate a correlation hierarchy truncated rigorously at each hierarchical level in the thermodynamic limit of the fluid regime this truncation provides a specific invariant structure of multi particle densities of the plasma formulated on an infinite plane

on polynomials in three variables annihilated by two locally nilpotent derivations let b be a polynomial ring in three variables over an algebraically closed field k of characteristic zero we are interested in irreducible polynomials f b satisfying the following condition there exist nonzero locally nilpotent derivations d1 d2 b b such that ker d1 ker d2 and d1 f 0 d2 f the main result asserts that a nonconstant polynomial f b satisfies the above requirement if and only if its generic fiber k f k f b is isomorphic as an algebra over the field k k f to k x y z xy z for some nonconstant z k z

poisson brackets and two generated subalgebras of rings of polynomials let a f x1 x2 xn be a ring of polynomials over a field f on the variables x1 x2 xn it is well known see for example 11 that the study of automorphisms of the algebra a is closely related with the description of its subalgebras by the theorem of p m cohn 4 a subalgebra of the algebra f x is free if and only if it is integrally closed the theorem of a zaks 13 says that the dedekind subalgebras of the algebra a are rings of polynomials in a single variable a nowicki and m nagata 8 proved that the kernel of any nontrivial derivation of the algebra f x y char f 0 is also a ring of polynomials in a single generator an original solution of the occurrence problem for the algebra a using the groebner basis was given by d shannon and m sweedler 9 however the method of the groebner basis does not give any information about the structure of concrete subalgebras recall that the solubility of the occurrence problem for rings of polynomials over fields of characteristic 0 was proved earlier by g noskov 7 the present paper is devoted to the investigation of the structure of twogenerated subalgebras of a in the sequel we always assume that f is an arbitrary field of characteristic 0 let us denote by f the highest homogeneous part of an element f a and by f1 f2 fk the subalgebra of a generated by the elements f1 f2 fk a definition 1 a pair of polynomials f1 f2 a is called reduced if they satisfy the following conditions 1 f1 f2 are algebraically dependent 2 f1 f2 are algebraically independent 3 f1 f2 f2 f1 recall that a pair f1 f2 with condition 3 is usually called reduced condition 1 means that we exclude the trivial case when f1 f2 are algebraically independent we do not consider the case when f1 f2 are algebraically dependent concerning this case recall the well known theorem of s s abhyankar and t t moh 1 which says that if f g f x and f g f x then f or f

stable tameness of coordinates in two variables abstract given a commutative ring r with unity we prove in this paper that every coordinate of r 2 is stably tame

an asymptotic representation for the riemann zeta function on the critical line a representation for the riemann zeta function zeta s is given as an absolutely convergent expansion involving incomplete gamma functions which is valid for all finite complex values of s neq 1 it is then shown how use of the uniform asymptotics of the incomplete gamma function leads to a new asymptotic representation for zeta s on the critical line s frac 1 2 it when t rightarrow infty this new result involves an error function smoothing of an infinite sum and consequently shares some similarity to though is quite different from the recent asymptotic expansion for zeta frac 1 2 it developed by berry keating numerical examples suggest that term for term with a little extra computational effort the new representation is at least as accurate as the riemann siegel formula

the structure of a laurent polynomial fibration in n variables abstract bass connell and wright have proved that any finitely presented locally polynomial algebra in n variables over an integral domain r is isomorphic to the symmetric algebra of a finitely generated projective r module of rank n in this paper we prove a corresponding structure theorem for a ring a which is a locally laurent polynomial algebra in n variables over an integral domain r viz we show that a is isomorphic to an r algebra of the form sym r q i 1 where q is a direct sum of n finitely generated projective r modules of rank one and i is a suitable invertible ideal of the symmetric algebra sym r q further we show that any faithfully flat algebra over a noetherian normal domain r whose generic and codimension one fibres are laurent polynomial algebras in n variables is a locally laurent polynomial algebra in n variables over r

generalized nonlinear fluctuation dissipation relation for the one component plasma we present a novel procedure leading to compact and tractable cubic p 3 and quartic p 4 scalar fluctuation dissipation relations for the classical one component plasma subjected to a weak external coulombic perturbation there is no external magnetic field the ultimate goal here is to establish a generalized nonlinear fluctuation dissipation relation in which a single p 1 point dynamical structure function is expressed as a linear combination of pth order external density response functions when cast in this form such an architecture by definition is invariant with respect to rotation on the p 1 sided polygon formed by the wave vector frequency arguments of the dynamical structure function the application of this rotation symmetry to suppress unphysical isolated singularities is a basic ingredient of the derivation

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

on double danielewski surfaces and the cancellation problem abstract we study a two dimensional family of affine surfaces which are counter examples to the cancellation problem we describe the makar limanov invariant of these surfaces determine their isomorphism classes and characterize the automorphisms of these surfaces

superoperator representation of nonlinear response unifying quantum field and mode coupling theories computing response functions by following the time evolution of superoperators in liouville space whose vectors are ordinary hilbert space operators offers an attractive alternative to the diagrammatic perturbative expansion of many body equilibrium and nonequilibrium green s functions the bookkeeping of time ordering is naturally maintained in real physical time allowing the formulation of wick s theorem for superoperators giving a factorization of higher order response functions in terms of two fundamental green s functions backward propagations and analytic continuations using artificial times keldysh loops and matsubara contours are avoided a generating functional for nonlinear response functions unifies quantum field theory and the classical mode coupling formalism of nonlinear hydrodynamics and may be used for semiclassical expansions classical response functions are obtained without the explicit computation of stability matrices

seminormality projective algebras and invertible algebras unlike module theory not much is known about the projective objects in the category of commutative r algebras where r is a fixed commutative ground ring the similarity between projective r modules and projective ralgebras does not extend much farther to our knowledge than to the following simple facts the free commutative r algebras namely r ti is t are projective and all projectives are retracts of free objects indeed this characterizes the projective r algebras in the definition of projectivity for algebras one must be sure to specify which epimorphisms one wants to admit if all epimorphisms of r alg the category of commutative r algebras are admitted one can show that only r itself is projective it is very reasonable to admit just the surjections which is what we are doing examples nearly as obvious as the free algebras can be found by taking the symmetric r algebra s r over a projective r module p and this algebra is obviously projective but free only if p is a free r module 3 lemma 4 61 besides these there are not many known examples abhyankar eakin and heinzer 1 4 1 and costa 5 3 5 proved some affirmative results if r is a field or more generally a ufd then every projective r algebra in one variable is a symmetric algebra of some rmodule p which must be a rank one projective module in this paper we call a domain a over a domain r an algebra in one variable over r if a is of finite type over r and quot a has transcendency degree 1 over quot r for our definition in case r is not a domain see definition 1 11 below as to the case of two variables we can say something for r a perfect field due to the recent general solution of the cancellation problem any projective kalgebra a of finite type over the perfect field k with tr deg a 2 is isomorphic to k x y for a proof we refer to 10 theorems 3 and 41 a slight change is needed in that author s theorem 4 we do not go into the details here

a triviality criterion for a2 fibrations over a ring containing q let r be an arbitrary commutative ring with unity containing q and let a be a stably trivial a2 fibration over r in this paper we prove that a is trivial if and only if it has a fixed point free locally nilpotent r derivation

zariski cancellation problem for non domain noncommutative algebras we study zariski cancellation problem for noncommutative algebras that are not necessarily domains

an algebraic proof of a cancellation theorem for surfaces let k be an algebraically closed field of arbitrary characteristic we give a self contained algebraic proof of the following statement if v is an affine surface over k such that v k k3 then v k2 this fact which is due to fujita miyanishi sugie and russell solves the zariski cancellation problem for surfaces to achieve our proof we first show that if a is a finitely generated domain with ak a a then ak a x a

a stable recurrence for the incomplete gamma function with imaginary second argument even though the two term recurrence relation satisfied by the incomplete gamma function is asymptotically stable in at least one direction for an imaginary second argument there can be a considerable loss of correct digits before stability sets in we present an approach to compute the recurrence relation to full precision also for small values of the arguments when the first argument is negative and the second one is purely imaginary a detailed analysis shows that this approach works well for all values considered

absolutely integral homomorphisms abstract absolutely integral algebras over a field are the subject of a chapter in field theory they are also known as geometrically integral algebras in algebraic geometry we generalize this concept to arbitrary commutative algebras our first main result is that a ring homomorphism is absolutely integral if and only if it becomes flat after an integral base change and its fibers are geometrically integral these homomorphisms are characterized with the help of absolute integral closedness property and do have some transcendence properties they generally preserve normality regularity and the complete intersection property

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

a note on retracts of polynomial rings in three variables abstract for retracts of the polynomial ring in 1 costa asks us whether every retract of k n is also the polynomial ring or not where k is a field in this paper we give an affirmative answer in the case where k is a field of characteristic zero and n 3

on the cancellation problem for the affine space mathbb a 3 in characteristic p we show that the cancellation conjecture does not hold for the affine space mathbb a 3 k over any field k of positive characteristic we prove that an example of t asanuma provides a three dimensional k algebra a for which a is not isomorphic to k x1 x2 x3 although a t is isomorphic to k x1 x2 x3 x4

a new proof of the stillinger lovett complete shielding condition it is shown that any equilibrium state of classical charged particles with correlation having a spatial decay faster than 1 x v 2 in dimensionv 2 3 obeys the stillinger lovett second moment condition under the same clustering hypothesis arbitrary localized external charge distributions are completely shielded