the realization of hyperelliptic curves through endomorphisms of kronecker modules abstract let k be an algebraically closed field and a the kronecker algebra over k a general problem is to study the endomorphism algebras of a modules m that are extensions of finite dimensional torsion free rank one a modules p by infinite dimensional torsion free rank one a modules n such endomorphism algebras can be studied by means of a quadratic polynomial f y in one variable y over the rational function field k x we call this f y the regulator of the extension we prove that if the regulator has non zero discriminant then end m is a noetherian commutative k algebra we also prove that subject to a regulator with non zero discriminant end m is affine over k if and only if end n is affine in which case end m is the coordinate ring of a hyperelliptic curve

primes in short intervals contrary to what would be predicted on the basis of cramer s model con cerning the distribution of prime numbers we develop evidence that the distribution of x h x for 0 x n is approximately normal with mean h and variance h log n h when n h n 1

castelnuovo mumford regularity of simplicial toric rings abstract bounds for the castelnuovo mumford regularity of simplicial toric rings are given which are close to the bound stated in eisenbud goto s conjecture

complex architecture of primes and natural numbers prime numbers and their distribution in the sequence of natural numbers are of interest for fundamental as well as practical reasons the authors introduce a stochastic model based on a network representation that closely matches some statistical properties of the prime numbers and that can be useful as a tool in the study of the distribution of primes

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

commutative algebras of rational function matrices as endomorphisms of kronecker modules ii this paper is the sequel to the paper commutative algebras of rational function matrices as endomorphisms of kronecker modules i here we focus on those kronecker modules that are rank 2 extensions of finite dimensional torsion free modules by the unique generic module r we show that if such an extension is indecomposable then its endomorphism algebra is trivial this resolves the missing case from the previous paper where the effects of height functions on the endomorphism algebras of extensions are studied

operator dimensions and surface exponents for the non linear schrodinger model at t 0 using the euler maclaurin formula the authors compute finite size corrections to the ground and excited state energies and momenta this enables them to obtain all possible operator scaling dimensions at the critical point t 0 and surface exponents for a variety of boundary conditions they extend the predictions of conformal invariance to include green functions with oscillating terms

a determinant representation for a correlation function of the scaling lee yang model we consider the scaling lee yang model it corresponds to the unique perturbation of the minimal cft model this model is not unitary we are using an expression for form factors in terms of symmetric polynomials in order to obtain a closed expression for the correlation function of the trace of the energy momentum tensor this expression is a determinant of an integral operator similar determinant representations were proven to be useful not only for quantum correlation functions but also in matrix models

bass numbers in the graded case a invariant formulas and an analogue of faltings annihilator theorem abstract let r n 0rn be a positively graded commutative noetherian ring a graded r module is called indecomposable respectively injective if it is indecomposable respectively injective in the category of graded r modules a graded prime ideal of r is called irrelevant if it contains r n 0rn let m n z mn be a finitely generated graded r module the first main result of the paper presents a refinement of the theory of the bass numbers of m with respect to an irrelevant graded prime ideal p of r this refinement is related to the siting in terms of degrees of indecomposable injective direct summands having associated prime p of terms in the minimal injective resolution of m the second main result has a consequence which can be described in terms of the castelnuovo regularity r of m any indecomposable injective direct summand having irrelevant associated prime of any term in the minimal injective resolution of m must vanish in all degrees greater than r results of this type provide uniform upper bounds for the degrees of influence of such irrelevant primes and these bounds are exploited in the later sections of the paper to a generalize known a invariant formulas due to e hyry and n v trung b to throw new light on the fact that for an ideal c 0 of r0 the sequence grademn c 0 n is ultimately constant and c to describe the ultimate constant value of that sequence by means of an analogue of faltings annihilator theorem for local cohomology

minimal free resolutions and asymptotic behavior of multigraded regularity let s be a standard nk graded polynomial ring over a field k let i be a multigraded homogeneous ideal of s and let m be a finitely generated zk graded s module we prove that the resolution regularity a multigraded variant of castelnuovo mumford regularity of inm is asymptotically a linear function this shows that the well known z graded phenomenon carries to the multigraded situation

the classification problem for torsion free abelian groups of finite rank in 1937 baer 5 introduced the notion of the type of an element in a torsion free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion free abelian groups of rank 1 since then despite the efforts of such mathematicians as kurosh 23 and malcev 25 no satisfactory system of complete invariants has been found for the torsion free abelian groups of finite rank n 2 so it is natural to ask whether the classification problem is genuinely more difficult for the groups of rank n 2 of course if we wish to show that the classification problem for the groups of rank n 2 is intractible it is not enough merely to prove that there are 2 such groups up to isomorphism for there are 2 pairwise nonisomorphic groups of rank 1 and we have already pointed out that baer has given a satisfactory classification for this class of groups in this paper following friedman stanley 11 and hjorth kechris 15 we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n 2 recall that up to isomorphism the torsion free abelian groups of rank n are exactly the additive subgroups of the n dimensional vector space qf which contain n linearly independent elements thus the collection of torsion free abelian groups of rank 1 r n can be naturally identified with the set s qn of all nontrivial additive subgroups of qn notice that s qn is a borel subset of the polish space p qn of all subsets of qn and hence s qn can be regarded as a standard borel space i e a polish space equipped with its associated o algebra of borel subsets here we are identifying p qn with the space 2 qn of all functions h qn 0 1 equipped with the product topology furthermore the natural action of gln q on the vector space qn induces a corresponding borel action on s qn and it is easily checked that if a b c s qn then a v b iff there exists an element f e gln q such that p a b it follows that the isomorphism relation on s qn is a countable borel equivalence relation if x is a standard borel space then a borel equivalence relation on x is an equivalence relation e c x2 which is a borel subset of x2 the borel equivalence relation e is said to be countable iff every e equivalence class is countable

the largest non vanishing degree of graded local cohomology modules abstract this paper gives characterizations for the largest non vanishing degree of the local cohomology modules of a graded ring s in terms of a reduction of s and of the associated graded ring of an ideal i in terms of any reduction of i as a consequence this invariant can be computed explicitly for the associated graded ring of any ideal generated by a d sequence and of ideals having small analytic deviation

asymptotic behaviour of arithmetically cohen macaulay blow ups this paper addresses problems related to the existence of arithmetic macaulayfications of projective schemes let y be the blow up of a projective scheme x projr along the ideal sheaf of i r it is known that there are embeddings y projk i e c for c d i e 1 where d i denotes the maximal generating degree of i and that there exists a cohen macaulay ring of the form k i e c if and only if h 0 y oy k h i y oy 0 for i 1 dimy 1 y is equidimensional and cohen macaulay cutkosky and herzog asked when there is a linear bound on c and e ensuring that k i e c is a cohen macaulay ring we obtain a surprising compelte answer to this question namely that under the above conditions there are well determined invariants and e0 such that k i e c is cohen macaulay for all c d i e and e e0 our approach is based on recent results on the asymptotic linearity of the castelnuovo mumford regularity of ideal powers we also investigate the existence of a cohen macaulay rees algebra of the form r i e ct which provides an arithmetic macaulayfication for x if r has negative a invariant we prove that such a cohen macaulay rees algebra exists if and only if oy ox r i oy 0 for i 0 y is equidimensional and cohen macaulay especially these conditions imply the cohen macaulayness of r i e ct for all c d i e and e e0 the above results can be applied to obtain several new classes of cohen macaulay algebras

gaussian unitary ensemble eigenvalues and riemann ζ function zeros a nonlinear equation for a new statistic for infinite gaussian unitary ensemble random matrices the probability density function s mathrm nn t for the nearest neighbor eignenvalue spacing as distinct from the spacing between consecutive eigenvalues is computed in terms of the solution of a certain nonlinear equation which generalizes the ensuremath sigma form of the painlev e v equation comparison is made with the empirical value of s mathrm nn t for the zeros of the riemann ensuremath zeta function on the critical line including data from 10 6 consecutive zeros near zero number 10 20

supporting degrees of multi graded local cohomology modules for a finitely generated graded module m over a positively graded commutative noetherian ring r the second author established in 1999 some restrictions which can be formulated in terms of the castelnuovo regularity of m or the so called a invariant of m on the supporting degrees of a graded indecomposable graded injective direct summand with associated prime ideal containing the irrelevant ideal of r of any term in the minimal graded injective resolution of m earlier in 1995 t marley had established connections between finitely graded local cohomology modules of m and local behaviour of m across proj r the purpose of this paper is to present some multi graded analogues of the above mentioned work

the classification problem for s local torsion free abelian groups of finite rank abstract suppose that n 2 and that s t are sets of primes then the classification problem for the s local torsion free abelian groups of rank n is borel reducible to the classification problem for the t local torsion free abelian groups of rank n if and only if s t

determinant representation for dynamical correlation functions of the quantum nonlinear schrodinger equation painleve analysis of correlation functions of the impenetrable bose gas by m jimbo t miwa y mori and m sato 1 was based on the determinant representation of these correlation functions obtained by a lenard 2 the impenetrable bose gas is the free fermionic case of the quantum nonlinear schrodinger equation in this paper we generalize the lenard determinant representation for langle psi 0 0 psi dagger x t rangle to the non free fermionic case we also include time and temeprature dependence in forthcoming publications we shall perform the jmms analysis of this correlationl function this will give us a completely integrable equation and asymptotic for the quantum correlation function of interacting fermions

applications of linear functionals to kronecker modules ii this paper continues the investigation of a class of modules er 2 constructed from submodules of the module of rational functions k x over the polynomial ring k x given by a height function h sequences of linear functionals l l2 lr on k x and a sequence of positive integers m2 mr the emphasis is on when 1 card k h to each subset l of l2 lr is attached a field extension kl of the prime subfield of k let ml1 and ml2 be modules constructed from the sequences of linear functionals l1 and l2 if ml1 is isomorphic to ml2 then kl1 kl2 are contained in finitely generated field extensions of each other let li l li i 2 r if ml is not purely simple then for some i in 2 r kli is contained in a finitely generated extension of kli these results are then used to obtain the following theorems assuming that 1 supp h then there are exactly 0 isomorphism classes of completely decomposable modules in er there are at least c isomorphism classes of i indecomposable but not purely simple modules in er ii purely simple modules in er c the cardinality of the continuum

proofs of two conjectures related to the thermodynamic bethe ansatz we prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic bethe ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel frac e u theta u theta prime cosh frac theta theta prime 2

dual field formulation of quantum integrable models equal time correlators are studied in completely integrable models the main example is the quantum non linear schrodinger equation introduction of an auxiliary fock space permits us to represent the generating functional of correlators in the form of a determinant of the integral operator

correlations for the cauchy and generalized circular ensembles with orthogonal and symplectic symmetry the generalized circular ensemble which specifies a spectrum singularity in random matrix theory is equivalent to the cauchy ensemble via a stereographic projection the cauchy weight function is classical and as such the n point distribution function in the cases of orthogonal and symplectic symmetry have expressions in terms of quaternion determinants with elements given in an explicit form suitable for asymptotic analysis the asymptotic analysis is undertaken in the neighbourhood of the spectrum singularity in both cases and it is shown that each quaternion determinant is specified by a single function involving bessel functions

determinant representation for correlation functions of spin 1 2 xxx and xxz heisenberg magnets we consider zero temperature correlation functions of the spin 1 2 xxz heisenberg chain in the critical regime 1 1 in a magnetic field starting from the algebraic bethe ansatz we derive representations for various correlation functions in term of determinats of fredholm integral operators

a criterion for pure simplicity abstract every purely simple kronecker module m of finite rank is an extension of a finite dimensional torsion free module n by an infinite dimensional torsion free module f of rank 1 but not conversely we prove that for each such n and f a purely simple extension m does exist extensions of n by f are built up using linear functionals on k x the space of rational functions our existence proof is based on a criterion in terms of such functionals for the pure simplicity of an extension m the possibility of embedding such m back into modules of rank 1 is also explored

the determinant representation for quantum correlation functions of the sinh gordon model we consider the quantum sinh gordon model in this paper form factors in this model were calculated by mussardo and colleagues we sum up all contributions of form factors and obtain a closed expression for a correlation function this expression is a determinant of an integral operator similar determinant representations have been proven to be useful not only in the theory of correlation functions but also in matrix models

curves arising from kronecker modules abstract let k be an algebraically closed field a kronecker module m is a pair of k vector spaces s t together with a k bilinear map k 2 s t the space s is called the domain space of m while t is called the range space of m to each power series in k x we attach a kronecker module p whose domain and range spaces are denoted by v and v respectively both v and v are modules over the endomorphism algebra end p of p we show that if end p is non trivial then the sequence of coefficients of is defined by a linear or a quadratic recursion in the quadratic case end p is the coordinate ring of an affine curve an affine curve is called realizable when its coordinate ring is isomorphic to some end p we show that the realizable curves can be constructed up to birational equivalence by pairs of non zero polynomials p q with deg q p the curves realized herein are embedded in k d where d deg p the planarity of these curves when d 3 remains open our main result on realization of curves is that a cubic curve is realizable if and only if its coordinate ring is either k x or the coordinate ring of a cubic in weierstrass form

linear algebraic groups and countable borel equivalence relations this paper is a contribution to the study of borel equivalence relations on standard borel spaces i e polish spaces equipped with their borel structure in mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants frequently these objects can be viewed as elements of a standard borel space x and the equivalence turns out to be a borel equivalence relation e on x a complete classification of x up to e consists of finding a set of invariants i and a map c x i such that xey c x c y for this to be of any interest both i and c must be explicit or definable and as simple and concrete as possible the theory of borel equivalence relations studies the set theoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems

measure reducibility of countable borel equivalence relations we show that every basis for the countable borel equivalence relations strictly above mathbb e 0 under measure reducibility is uncountable thereby ruling out natural generalizations of the glimm effros dichotomy we also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base using arguments substantially simpler than those previously employed

a family of commutative endomorphism algebras abstract let k x be the quotient field of the polynomial ring k x over an algebraically closed field k given an integer n 2 a finite sequence m m 2 m n of n 1 positive integers a sequence 2 n of k linear functionals on k x and a height function h there is attached a kronecker module e e m h in this paper we show that when e has no finite dimensional direct summand then end e is a commutative k subalgebra of the algebra of n n matrices over k x consequently when end e is an integral domain its transcendence degree over k is at most one we give sufficient conditions for end e to be isomorphic to k and also show that k is the only field that occurs as end e examples where k x k x x 1 and some subalgebras of k x n occur as end e are given the determination of all the commutative k subalgebras of m n k x realizable as end e remains open yet surprisingly it can be shown that the coordinate ring of a cubic equation in weierstrass normal form y 2 x 3 ax 2 bx c is realizable

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

associated primes of graded components of local cohomology modules the i th local cohomology module of a nitely generated graded module m over a standard positively graded commutative noetherian ring r with respect to the irrelevant ideal r is itself graded all its graded components are nitely generated modules over r0 the component of r of degree 0 it is known that the n th component hi r m n of this local cohomology module hi r m is zero for all n 0 this paper is concerned with the asymptotic behaviour of assr 0 hi r m n as n the smallest i for which such study is interesting is the niteness dimension f of m relative to r de ned as the least integer j for which hj r m is not nitely generated brodmann and hellus have shown that assr 0 h f r m n is constant for all n 0 that is in their terminology assr 0 h f r m n is asymptotically stable for n the rst main aim of this paper is to identify the ultimate constant value under the mild assumption that r is a homomorphic image of a regular ring our answer is precisely the set of contractions to r0 of certain relevant primes of r whose existence is con rmed by grothendieck s finiteness theorem for local cohomology brodmann and hellus raised various questions about such asymptotic behaviour when i f they noted that singh s study of a particular example in which f 2 shows that assr 0 h3 r r n need not be asymptotically stable for n the second main aim of this paper is to determine for singh s example assr 0 h3 r r n quite precisely for every integer n and thereby answer one of the questions raised by brodmann and hellus

completeness of the isomorphism problem for separable c algebras this paper studies the descriptive set theoretical complexity of the isomorphism problem for separable c algebras we prove that the isomorphism problem for separable simple ai c algebras is complete in the class of orbit equivalence relations this means that any isomorphism problem arising from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple separable ai c algebras

finite size effects in conformal field theories and non local operators in one dimensional quantum systems the authors generalise the well known connection between critical exponents and finite size effects in conformal field theories to non local operators and find the long wave asymptotics of vacuum expectation values of some non local operators in one dimensional quantum systems

two point correlation functions in perturbed minimal models two point correlation functions of the off critical primary fields are considered in the perturbed minimal models they are given as infinite series of form factor contributions the form factors of are conjectured from the known results for those of and the conjectured form factors are rewritten in a form which is convenient for summation

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

multigraded regularity a invariant and the minimal free resolution abstract in recent years two different multigraded variants of castelnuovo mumford regularity have been developed namely multigraded regularity defined by the vanishing of multigraded pieces of local cohomology modules and the resolution regularity vector defined by the multi degrees in a minimal free resolution in this paper we study the relationship between multigraded regularity and the resolution regularity vector our method is to investigate multigraded variants of the usual a invariant this in particular provides an effective approach to examining the vanishing of multigraded pieces of local cohomology modules with respect to different multigraded ideals

on the complexity of the isomorphism relation for finitely generated groups working within the framework of descriptive set theory we show that the isomorphism relation for finitely generated groups is a universal essentially countable borel equivalence relation we also prove the corresponding result for the conjugacy relation for subgroups of the free group on two generators the proofs are group theoretic and we refer to descriptive set theory only for the relevant definitions and for motivation for the results

the quantum inverse scattering method approach to correlation functions the inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with ther matrix of xxx type these models include the one dimensional bose gas and the heisenberg xxx model the algebraic questions of the problem are considered

temperature correlators of the impenetrable bose gas as an integrable system it is shown that the temperature equal time correlators of impenetrable bosons in one space dimension are described by a classical integrable system partial differential equations for two point as well as for multipoint correlators are obtained the short distance and low density expansions are constructed

the isomorphism problem for torsion free abelian groups is analytic complete we prove that the isomorphism problem for torsion free abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy namely 1 complete

determinant representation for a quantum correlation function of the lattice sine gordon model we consider a completely integrable lattice regularization of the sine gordon model with discrete space and continuous time we derive a determinant representation for a correlation function which in the continuum limit turns into the correlation function of local fields the determinant is then embedded into a system of integrable integro differential equations the leading asymptotic behaviour of the correlation function is described in terms of the solution of a riemann hilbert problem rhp related to the system of integro differential equations the leading term in the asymptotical decomposition of the solution of the rhp is obtained

fredholm determinants and the mkdv sinh gordon hierarchies for a particular class of integral operatorsk we show that the quantity phi log det i k log det i k satisfies both the integrated mkdv hierarchy and the sinh gordon hierarchy this proves a conjecture of zamolodchikov