the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

a new criterion for the first case of fermat s last theorem a new criterion of the first case of fermat last theorem is derived this criterion is based on special sums of reciprocals

quantum field theory grassmannians and algebraic curves this paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite grassmannians and in part to pointing out the existence of a close analogy between conformal field theory on riemann surfaces and the modern theory of automorphic representations along the way we develop a multiplicative analog of the usual additive ward identities of current algebra we also reformulate the additive ward identities in a way which may be useful in terms of the residues of operator valued differential forms a concluding section is devoted to some remarks on string field theory in an appendix we attempt to clarify the recent construction by beilinson manin and schechtman of what might be called global virasoro algebras

the q division ring and its fixed rings abstract we prove that the fixed ring of the q division ring k q x y under any finite group of monomial automorphisms is isomorphic to k q x y for the same q in a similar manner we also show that this phenomenon extends to an automorphism that is defined only on k q x y and does not restrict to k q x 1 y 1 we then use these results to answer several questions posed by artamonov and cohn about the endomorphisms and automorphisms of k q x y

global and krull dimensions of quantum weyl algebras abstract given a hecke symmetry r a giaquinto and j zhang defined a natural quantization an r of the nth weyl algebra an based on r and studied many ring theoretic properties of rings a2 ja b arising from the jordan hecke symmetry and an q pi j arising from the standard multiparameter hecke symmetry here we compute the global and krull dimensions in the cases that were left open namely we show that over any field k of characteristic zero gldim a2 ja b kdim a2 ja b 3 for any a b k with a b and gldim an 1 pi j kdim an 1 pi j n

zero sets of functions and hidden hierarchies of kdv type the zero sets of kdv functions are characterized in terms of the stratification of the infinite grassmannian it is shown that these sets are related to integrable hierarchies arising from schrodinger equations with energy dependent potentials

regular modules over 2 dimensional quantum beilinson algebras of type s in the study of a finite dimensional hereditary algebra of infinite representation type understanding regular modules is essential recently herschend iyama and oppermann introduced the notions of d representation infinite algebra and d regular module extending the above notions to finite dimensional algebras of global dimension d ge 1 since the beilinson algebras of as regular algebras of dimension d 1 are typical examples of d representation infinite algebras the purpose of this paper is to study the behavior of d regular modules over such algebras in particular we will show that the isomorphism classes of simple 2 regular modules over a 2 representation tame quantum beilinson algebra of type s are parameterized by mathbb p 2

the non local delta problem and 2 1 dimensional soliton equations a method of constructing 2 1 dimensional non linear integrable equations and their solutions by means of the non local delta problem is developed a basic set of equations is obtained by using different normalisations of the non local delta problem and the lagrangian of the set is found other integrable equations which are degenerate cases of the basic set are also lagrangian

automorphisms and isomorphisms of quantum generalized weyl algebras abstract we classify up to isomorphism the quantum generalized weyl algebras and determine their automorphism groups in all cases in a uniform way including the case where the parameter q is a root of unity thereby completing the results obtained by bavula and jordan 2001 5 and richard and solotar 2006 12

matrix bipolar asymptotic modules for solving 2 1 dimensional non linear evolution equations with constraints an infinite hierarchy of solvable systems of purely differential non linear equations is introduced within the framework of asymptotic modules each system consists of 2 1 dimensional evolution equations for at most four complex functions and of quite strong differential constraints it may be interpreted formally as an integro differential equation in 1 1 dimensions

on fermat s quotient base two this paper extends the search for solutions of the congruence 2p1 0 modp2 to the limit p 6 109 no solution except the well known p 1093 and p 3511 was found in 1969 brillhart tonascia and weinberger 1 reported on the search for solutions of the congruence a 1 modp2 a 2 1 199 for a 2 a special effort was made to consider all p 3 109 only the two known solutions p 1093 and p 3511 were found having the occasional opportunity to use the illiac iv one of the projects decided upon was the recalculation and extension of the above result for a 2 by a somewhat different method the calculation was pushed to twice the above limit that is top 6 109 without finding any further solutions the parallel construction of the illiac iv makes it possible to look for 64 different values of p at the same time the speed of illiac iv is such that a range of 100000 numbers can be searched in one second thus the range for p 6 109 was broken up into 60 runs of 1000 seconds each the program was run as one of a few backlog problems over the past two years let n belong to one of the 64 residue classes that are prime to 240 and let 2m am nbm mod n2 where 0 am n 0 bm n for numbers n as large as 109 the number n2 is a doubly precise integer nevertheless the calculation of am and bm can be accomplished by single precision arithmetic in only o log m operations in fact one uses one or the other of the following two recurrences if m 2h 1 then am 2a2h and bm 2b2h mod n if m 2h and if a 2 rm nqm 0 rm n and if 2ahbh dm mod n then am rm and bm qm dm received march 30 1980 1980 mathematics subject classification primary 10 04 1981 american mathematical society 0025 5718 81 0000 0027 0 1 50 289 this content downloaded from 157 55 39 58 on tue 11 oct 2016 04 26 58 utc all use subject to http about jstor org terms

field of fractions of certain quantum algebras abstract in this paper we show that the field of fractions of certain algebras appearing in the theory of quantum groups are isomorphic to the field of fractions of quantum n space when the parameter q is not a root of 1 more precisely the quantum serre relations of type a n and the relations defining the quantum weyl algebra induce q communtation relations in the field of fractions also we study the problem of separation of these skew fields using the standard embeddings in non commutative series

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

invariant prime ideals in quantizations of nilpotent lie algebras de concini kac and procesi defined a family of subalgebras u of a quantized universal enveloping algebra uq g associated to the elements of the corresponding weyl group w they are deformations of the universal enveloping algebras u n adw n where n are the nilradicals of a pair of dual borel subalgebras based on results of gorelik and joseph and an interpretation of u as quantized algebras of functions on schubert cells we construct explicitly the h invariant prime ideals of each u and show that the corresponding poset is isomorphic to w where h is the group of group like elements of uq g moreover for each h prime of u w we construct a generating set in terms of demazure modules related to fundamental representations using results of ramanathan and kempf we prove similar theorems for vanishing ideals of closures of torus orbits of symplectic leaves of related poisson structures on schubert cells in flag varieties

a special extension of wieferich s criterion the following theorem is proved in this paper if the first case of fermat s last theorem does not hold for sufficiently large prime 1 then e x1 2 kl x k x i 0 modl1 x nn for all pairs of positive integers n k n 94 0 k ni the proof of this theorem is based on a recent paper of skula and uses computer techniques 0 introduction the first case of fermat s last theorem states that for each odd prime i the equation xi y1 zi 0 has no integral solution x y z with i t xyz one of many methods investigating this problem was introduced by a wieferich this method is connected with the fermat quotients ql a al l q a i defined for each integer a such that a is not divisible by 1 let us assume in this paragraph that is an odd prime which does not satisfy the first case of fermat s last theorem in 1909 wieferich 7 published the following important result q 2 0 mod1 many mathematicians have extended this wieferich criterion the latest result is due to a granville and b monagan 1 and states q1 p 0 mod 1 for each prime p such that p 89 these considerations have been generalized by l skula he studied the sums s k n e 1 2 kl x k 1 1 s k n ex x

new bound for the first case of fermat s last theorem we present an improvement to gunderson s function which gives a lower bound for the exponent in a possible counterexample to the first case of fermat s last theorem assuming that the generalized wieferich criterion is valid for the first n prime bases the new function increases beyond n 29 unlike gunderson s and it increases more swiftly using the recent extension of the wieferich criterion to n 24 by granville and monagan the first case of fermat s last theorem is proved for all prime exponents below 156 442 236 847 241 729

singular sector of the kadomtsev petviashvili hierarchy operators of nonzero index and associated integrable systems integrable hierarchies associated with the singular sector of the kadomtsev petviashvili kp hierarchy or equivalently with operators of nonzero index are studied they arise as the restriction of the standard kp hierarchy to submanifolds of finite codimension in the space of independent variables for higher index these hierarchies represent themselves as families of multidimensional equations with multidimensional constraints the dressing method is used to construct these hierarchies hidden korteweg de vries boussinesq and hidden gelfand dikii hierarchies are considered too

effacement des derivations et spectres premiers des algebres quantiques abstract given any commutative field k and any iterated ore extension r k x1 x2 2 2 xn n n satisfying some suitable assumptions we construct the so called derivative elimination algorithm it consists of a sequence of changes of variables inside the division ring f fract r starting with the indeterminates x1 xn and terminating with new variables t1 tn these new variables generate some quantum affine space r such that f fract r this algorithm induces a natural embedding spec r spec r which satisfies the following property if p is in spec r then fract r p fract r p we study both the derivative elimination algorithm and natural embedding and use them to produce for the general case a common proof of the quantum gelfand kirillov property for the prime homomorphic images of the following quantum algebras u q w g b q w g w w r q g where g denotes any complex semi simple connected simply connected lie group with associated lie algebra g and weyl group w quantum matrices algebras quantum weyl algebras and quantum euclidean respectively symplectic spaces another application will be given in g cauchon j algebra to appear in the general case the prime spectrum of any quantum matrices algebra satisfies the normal separation property

solution of a q difference noether problem and the quantum gelfand kirillov conjecture for mathfrak gl _n it is shown that the q difference noether problem for all classical weyl groups has a positive solution simultaneously generalizing well known results on multisymmetric functions of mattuck proc am math soc 19 764 765 1968 and miyata nagoya math j 41 69 73 1971 in the case q 1 and q deforming the noncommutative noether problem for the symmetric group futorny et al in adv math 223 773 796 2010 it is also shown that the quantum gelfand kirillov conjecture for mathfrak gl n for a generic q follows from the positive solution of the q difference noether problem for the weyl group of type d n the proof is based on the theory of galois rings futorny and ovsienko in j algebra 324 598 630 2010 from here we obtain a proof of the quantum gelfand kirillov conjecture for mathfrak gl n and for a certain extension of mathfrak sl n previously the case of mathfrak sl n was shown by fauquant millet j algebra 218 93 116 1999 and by alev and dumas j algebra 170 229 265 1994 for n 2 3 moreover we give an explicit description of the skew fields of fractions for u q mathfrak gl n and u q mathrm ext mathfrak sl n which generalizes the results of alev and dumas j algebra 170 229 265 1994

a solvable hierarchy of n 1 dimensional non linear evolution equations with constraints matrix multipolar asymptotic modules are used for introducing and solving a hierarchy of systems of purely differential non linear equations each system consists of n 1 dimensional evolution equations and of quite strong differential constraints specific constructions of matrix multipolar asymptotic modules are given in particular delta constructions

algebra of derivations of lie algebras abstract we show a method to determine the space of derivations of any lie algebra and in particular we apply this method to a special class of lie algebras those nilpotent with low nilindex most calculations have been supported by the software mathematica 3 0

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

quantification de la localisation de dixmier de u sln 1 c resume in this article we extend to the quantum case the classical result of dixmier according to which up to localization the enveloping algebra of sl n 1 c is a central polynomial extension of the enveloping algebra of some parabolic subalgebra in fact the latter has a semiinvariant element d by which it is sufficient to localize to obtain the result the methods we use here are quite different from those used by dixmier indeed in the quantum case the powers of the indeterminate q would make calculations similar to those of dixmier much too complicated on the other hand when we use the rosso form which allows us to work in the hopf dual of q sl n 1 c the calculations become much easier than they would be in q sl n 1 c

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

on theq commutations inuq n at roots of one abstract let e be a root of one and g a semisimple lie algebra with triangular decomposition g n h n letu e resp ures e be the nonrestricted resp restricted quantum enveloping algebra of n we prove that fract u eis a quantum weyl field we then give a description of the e center ofu e letufin ebe the finite part ofures e via the drinfeld correspondence theufin e covariant space of a weyl module is e central in case g s l n this enables us to describe this space in terms of semistandard young tableaux

limit matrices for the toda flow and periodic flags for loop groups sato s theory of infinite dimensional grassmannians has been applied to explain the geometry of the k p equation s djkm it has been used as a tool to study blow up behaviors and to regularize the solutions near the blow up a vm2 the point is that realizing the k p flow as a holomorphic flow of planes enables one to follow what happens to the limiting planes as the equation in the original bad coordinates blows up the blow up behaviors are characterized by the various strata the orbit of planes visits in the grassmannian in this paper such ideas are applied to the n periodic toda flow on periodic jacobi matrices which translates into a flow on the space of n periodic flags of planes in the grassmannians indeed here the n periodic toda flow amounts to n coupled kp equations with special interactions between time flows u t how such matrices blow up has been studied in fi f1 ha a vm1 for arbitrary lie algebras and kac moody lie algebras whereas this paper focusses on regularizing the flow near the blow up locus that is on finding the boundary of isospectral sets if n periodic jacobi matrices

pi degree parity in q skew polynomial rings abstract for k a field of arbitrary characteristic and r a k algebra we show that the pi degree of an iterated skew polynomial ring r x 1 1 1 x n n n agrees with the pi degree of r x 1 1 x n n when each i i satisfies a q i skew relation for q i k and extends to a higher q i skew i derivation we confirm the quantum gel fand kirillov conjecture for various quantized coordinate rings and calculate their pi degrees we extend these results to completely prime factor algebras

multiparameter twisted weyl algebras abstract we introduce a new family of twisted generalized weyl algebras called multiparameter twisted weyl algebras for which we parametrize all simple quotients of a certain kind both jordan s simple localization of the multiparameter quantized weyl algebra and hayashi s q analog of the weyl algebra are special cases of this construction we classify all simple weight modules over any multiparameter twisted weyl algebra extending results by benkart and ondrus we also describe all whittaker pairs up to isomorphism over a class of twisted generalized weyl algebras which includes the multiparameter twisted weyl algebras

the first case of fermat s last theorem is true for all prime exponents up to 714 591 416 091 389 we show that if the first case of fermat s last theorem is false for prime exponent p then p2 divides qp q for all primes q sq as a corollary we state the theorem of the title 1988 american mathematical society

coisotropic deformations of algebraic varieties and integrable systems coisotropic deformations of algebraic varieties are defined as those for which an ideal of the deformed variety is a poisson ideal it is shown that coisotropic deformations of sets of intersection points of plane quadrics cubics and space algebraic curves are governed in particular by the dkp wdvv dvn d2dtl equations and other integrable hydrodynamical type systems particular attention is paid to the study of two and three dimensional deformations of elliptic curves the problem of an appropriate choice of the poisson structure is discussed

a parametric family of subalgebras of the weyl algebra i structure and automorphisms an ore extension over a polynomial algebra f x is either a quantum plane a quantum weyl algebra or an infinite dimensional unital associative algebra ah generated by elements x y which satisfy yx xy h where h f x we investigate the family of algebras ah as h ranges over all the polynomials in f x when h 6 0 these algebras are subalgebras of the weyl algebra a1 and can be viewed as differential operators with polynomial coefficients we give an exact description of the automorphisms of ah over arbitrary fields f and describe the invariants in ah under the automorphisms we determine the center normal elements and height one prime ideals of ah localizations and ore sets for ah and the lie ideal ah ah we also show that ah cannot be realized as a generalized weyl algebra over f x except when h f in two sequels to this work we completely describe the derivations and irreducible modules of ah over any field

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

inverse and direct images for quantum weyl algebras in 18 wess and zumino gave a method for constructing noncommutative differential calculus or de rham complex on the quantum affine space associated to a hecke symmetry r also they constructed the corresponding algebra of linear differential operators since the algebra of linear differential operators on the n dimensional affine space is the n th weyl algebra this algebra is regarded as a quantum analogue of the weyl algebra and called the quantum weyl algebra associated to r let rq p be the multiparameter r matrix of the quantum deformation of gln parameterized by a scalar q and an n n matrix p pij in 3 for the quantum weyl algebra an q p associated to rq p demidov 6 and rigal 15 consider quantum versions of classical theory of the weyl algebras including bernstein s inequality and some ring theoretic properties of an q p have been studied in 1 2 9 10 11 etc in 11 jordan constructed a simple localization bn q p of an q p which is a better analogue of the weyl algebra an from the point of view of noncommutative ring theory the purpose of this paper is to define an analogue of the inverse and direct images for the quantum weyl algebra an q p and to investigate their properties in particular we prove a quantum analogue of kashiwara s theorem section 4 and consider preservation of holonomicity under inverse and direct images section 5 throughout this paper we fix a ground field k and let q be a nonzero element of k such that q is not a root of unity and we use the following q integer notation

algebras of invariant differential operators abstract we prove that an invariant subalgebra a n w of the weyl algebra a n is a galois order over an adequate commutative subalgebra when w is a two parameters irreducible unitary reflection group g m 1 n m 1 n 1 including the weyl group of type b n or alternating group a n or the product of n copies of a cyclic group of fixed finite order earlier this was established for the symmetric group in 15 in each of the cases above except for the alternating groups we show that a n w is free as a right left module similar results are established for the algebra of w invariant differential operators on the n dimensional torus where w is a symmetric group s n or orthogonal group of type b n or d n as an application of our technique we prove the quantum gelfand kirillov conjecture for u q s l 2 the first witten deformation and the woronowicz deformation

gunderson s function in fermat s last theorem we study gunderson s function which gives a bound on the first case of fermat s last theorem assuming that the generalized wieferich criterion is valid for the first n prime bases we note two unexpected phenomena

fields of fractions of quantum solvable algebras abstract we introduce the notion of pure q solvable algebra the quantum matrices quantum weyl algebra u q n are the examples it is proved that the skew field of fractions of a pure q solvable algebra r is isomorphic to the skew field of twisted rational functions this is a quantum version of the gelfand kirillov conjecture for solvable algebraic lie algebras

singular sector of the burgers hopf hierarchy and deformations of hyperelliptic curves we discuss the structure of shock singularities of the burgers hopf hierarchy it is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy the stratification is associated with the birkhoff decomposition of the grassmannian given by the set of linear spaces spanned by the hierarchy we then construct integrable hierarchy on each stratum and demonstrate that it describes a deformation of a hyperelliptic curve parametrizing the stratum the hierarchy is called the hidden burgers hopf hierarchy and we found the riemann invariant form and the hodograph solution

a quadratic poisson gel fand kirillov problem in prime characteristic the quadratic poisson gel fand kirillov problem asks whether the field of fractions of a poisson algebra is poisson birationally equivalent to a poisson affine space i e to a polynomial algebra k x1 xn with poisson bracket defined by xi xj ijxixj for some skew symmetric matrix ij mn k this problem was studied in 9 over a field of characteristic 0 by using a poisson version of the deleting derivation homomorphism of cauchon in this paper we study the quadratic poisson gel fand kirillov problem over a field of arbitrary characteristic in particular we prove that the quadratic poisson gel fand kirillov problem is satisfied for a large class of poisson algebras arising as semiclassical limits of quantised coordinate rings for we introduce the concept of higher poisson derivation which allows us to extend the poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic when a torus is acting rationally by poisson automorphisms on a poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring we prove under some technical assumptions that quotients by poisson prime torus invariant ideals also satisfy the quadratic poisson gel fand kirillov problem in particular we show that coordinate rings of determinantal varieties satisfy the quadratic poisson gel fand kirillov problem