behaviour of the frobenius map in a noncommutative world abstract we study various properties of the frobenius map x x p on a noncommutative algebra r over a field f of positive characteristic p we call r perfect whenever is surjective more generally we say that r is additively respectively multiplicatively p power closed if for every x y r there exists a positive integer n such that x p n y p n respectively x p n y p n is a p power in r if for example n were a ring homomorphism as when r is commutative then r would be additively and multiplicatively p power closed we show that a finite dimensional unital algebra r is lie nilpotent if and only if r is additively respectively multiplicatively p power closed and each of its simple homomorphic images has schur index 1 since not all finite dimensional perfect division algebras are lie nilpotent the schur index condition cannot be omitted we deduce that similar results hold for the class of all finitely generated pi algebras moreover for this same class we give a positive solution to the following problem reminiscent of the problems of kurosh and levitzki is every finitely generated perfect algebra finite dimensional

on artin s theorem and azumaya algebras a short proof of the existence of central polynomial of matrix rings is given and some of its applications the following characterization of azumaya algebras is proved r is azumaya iff there exist ai bi r i 1 k with ik 1 airbi z r r r and aibi 1 this is proved as a consequence of the following generalization of a theorem due to m artin and generalized by c procesi let r x1 xk be a p i ring a central noetherian subring then r is azumaya iff for every two sided ideal i in r z ri z r i z r where z r denotes the center of r

white noise driven ostrovsky equation abstract the current paper is devoted to the cauchy problem for the white noise driven ostrovsky equation with positive dispersion we obtain the local well posedness for the initial data u 0 h s r a e which is f 0 measurable with s 3 4 and l 2 0 s l 2 0 l 2 r h s 1 2 r and the globally well posedness for the initial data u 0 l 2 r a e which is f 0 measurable and l 2 0 0 l 2 0 l 2 r h 0 1 2 r the key ingredients that we used in this paper are some bilinear estimates in x s b type spaces given in subsection 2 1 ito formula the bdg inequality and stopping time techniques as well as frequency truncated technique our method can be applied to the case 0 which is just the kdv equation thus our result improves the local well posedness result of a de bouard a debussche and y tsutsumi 1999 3

stability of solitary waves and weak rotation limit for the ostrovsky equation abstract ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation in this model dispersion is taken into account while dissipation is neglected existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter which can be either positive or negative it is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations the issue of passing to the limit as the rotation parameter tends to zero for solutions of the cauchy problem is investigated on a bounded time interval

on gelfand kirillov dimension and related topics recall that in ii one always has gk m 2 max gk n gk e if equality holds for all short exact sequences of s modules then gk dimension is said to be exact for s modules in general exactness fails quite drastically even in situations which are otherwise considered to be well behaved for example g bergman 3 has constructed an afline pialgebra s having an ideal i of square 0 such that i is cyclic as right ideal of s yet gk s 3 gk s 1 2 thus quite stringent conditions have to be imposed on the algebra or the modules in question for equality to hold in ii on the other hand the integrality question i presumably has a positive answer for many classes of algebras and modules that are of interest although the gk dimension of an algebra module can be 0 1 co or any real number 2 2 0 co or any real number 2 1 warfield 17 it tends to be an integer or co in most cases which arise naturally thus no example of an affrne algebra s which is either noetherian or finitely presented but has finite non integral gk dimension seems to be known however positive results are rare some exceptions where gk dimension is known to be an integer or co are finitely presented monomial algebras govorov 9 almost commutative algebras tauvel 16 an noetherian pi algebras lorenz and small 131 gk dimension by its definition measures the rate of growth of the steps in certain canonically defined filtrations on algebras and modules often

analytical and numerical studies of weakly nonlocal solitary waves of the rotation modified korteweg de vries equation abstract a century ago the korteweg de vries kdv equation was derived as a model for weakly nonlinear long waves propagating down a channel when cross channel and depth variations are sufficiently weak in this article we study the steadily translating coherent structures of a generalization of this equation the rotation modified kdv equation which applies when coriolis forces are significant u t uu x u xxx x 2 u 0 where x is the down channel coordinate this is also called the ostrovosky equation the rmkdv solitary waves are weakly nonlocal due to the radiation of long waves where the radiation is proportional to the small parameter because of its simplicity the rmkdv coherent structures are the prototype of nonlocal solitary waves where the amplitude is a power of the small parameter micropterons we extend the matched asymptotic expansions of hunter to third order we also compute direct numerical solutions and bifurcations of the nonlinear eigenvalue problem new modes with multiple kdv like peaks flanked by small quasi sinusoidal oscillations are shown to be well described by matched asymptotics too

generalized quasilinearization method for mixed boundary value problems a generalized quasilinearization method for a nonlinear mixed bvp is developed and a sequence of approximate solutions converging monotonically and quadratically to a solution of the given problem is presented

linear equations over noncommutative graded rings abstract we call a graded connected algebra r effectively coherent if for every linear equation over r with homogeneous coefficients of degrees at most d the degrees of generators of its module of solutions are bounded by some function d d for commutative polynomial rings this property has been established by hermann in 1926 we establish the same property for several classes of noncommutative algebras including the most common class of rings in noncommutative projective geometry that is strongly noetherian rings which includes noetherian pi algebras and sklyanin algebras we extensively study so called universally coherent algebras that is such that the function d d is bounded by 2d for d 0 for example finitely presented monomial algebras belong to this class as well as many algebras with finite groebner basis of relations

rings with auslander dualizing complexes abstract a ring with an auslander dualizing complex is a generalization of an auslander gorenstein ring we show that many results which hold for auslander gorenstein rings also hold in the more general setting on the other hand we give criteria for existence of auslander dualizing complexes which show these occur quite frequently the most powerful tool we use is the local duality theorem for connected graded algebras over a field filtrations allow the transfer of results to nongraded algebras we also prove some results of a categorical nature most notably the functoriality of rigid dualizing complexes

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 dichotomy result for prime algebras of gelfand kirillov dimension two abstract let k be an uncountable field we show that a finitely generated prime goldie k algebra of quadratic growth is either primitive or satisfies a polynomial identity answering a question of small in the affirmative

symmetry and uniqueness of the solitary wave solution for the ostrovsky equation we consider herein the ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water or the capillary waves in a plasma with the effects of rotation using the modified sliding method we prove that the solitary wave moving to the left to the ostrovsky equation is symmetric about the origin and unique up to translations we also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed weak rotation and dispersive parameter

the cauchy problem for the ostrovsky equation with negative dispersion at the critical regularity abstract in this paper we investigate the cauchy problem for the ostrovsky equation x u t x 3 u 1 2 x u 2 u 0 in the sobolev space h 3 4 r here 0 0 corresponds to the positive negative dispersion of the media respectively p isaza and j mejia 2006 13 2009 15 k tsugawa 2009 26 proved that the problem is locally well posed in h s r when s 3 4 and ill posed when s 3 4 by using some modified bourgain spaces we prove that the problem is locally well posed in h 3 4 r with 0 and 0 the new ingredient that we introduce in this paper is lemma 2 1 lemma 2 2 lemma 2 3 lemma 2 4 lemma 2 5 lemma 2 6

global cauchy problem for the ostrovsky equation abstract in this article we consider the initial value problem for the ostrovsky equation t u x 3 u x 1 u u x u 0 x r t r u x 0 u 0 x with initial data in sobolev spaces h s r using bourgain spaces we prove that the problem is globally well posed for s 3 10 for both signs in the equation

local and global well posedness for the ostrovsky equation abstract we consider the initial value problem for 0 1 t u x 3 u x 1 u uu x 0 x t r where u is a real valued function and are real numbers such that 0 and x 1 f i 1 f this equation differs from korteweg de vries equation in a nonlocal term nevertheless we obtained local well posedness in x s f h s r x 1 f l 2 r s 3 4 using techniques developed in c e kenig g ponce l vega well posedness of the initial value problem for the korteweg de vries equation j amer math soc 4 1991 323 347 for the case 0 we also obtain a global result in x 1 using appropriate conservation laws

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

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

convergence of the ostrovsky equation to the ostrovsky hunter one abstract we consider the ostrovsky equation which contains nonlinear dispersive effects we prove that as the diffusion parameter tends to zero the solutions of the dispersive equation converge to discontinuous weak solutions of the ostrovsky hunter equation the proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the l p setting

ideal classes of three dimensional artin schelter regular algebras abstract we determine the possible hilbert functions of graded rank one torsion free modules over three dimensional artin schelter regular algebras it turns out that as in the commutative case they are related to castelnuovo functions from this we obtain an intrinsic proof that the space of torsion free rank one modules on a non commutative p 2 is connected a different proof of this fact based on deformation theoretic methods and the known commutative case has recently been given by nevins and stafford sklyanin algebras and hilbert schemes of points math ag 0310045 for the weyl algebra it was proved by wilson invent math 133 1 1998 1 41

a generalized quasilinearization method for second order nonlinear differential equations with nonlinear boundary conditions we consider the differential equation l u f u where l is a formally self adjoint second order differential expression and f is nonlinear with nonlinear boundary conditions under appropriate assumptions on l f and the boundary conditions existence of solutions is established using the method of lower and upper solutions a generalized quasilinearization method is then developed for this problem and we obtain two monotonic sequences of approximate solutions converging quadratically to a solution of the equation

on the polynomial representations of current algebras abstract we obtain a sufficient condition for a representation of a lie algebra of the form l g where g is a finite dimensional simple lie algebra over an algebraically closed field f of zero characteristic and is an associative commutative algebra over f to be polynomial as a corollary we classify integrable graded irreducible modules over an extended loop algebra with finite dimensional homogeneous spaces the criteria is applied also to irreducible conformal modules over current algebras

well posedness and weak rotation limit for the ostrovsky equation we consider the cauchy problem of the ostrovsky equation we first prove the time local well posedness in the anisotropic sobolev space hs a with s a 2 3 4 and 0 a 1 by the fourier restriction norm method this result include the time local well posedness in hs with s 3 4 for both positive and negative dissipation namely for both 0 and 0 we next consider the weak rotation limit we prove that the solution of the ostrovsky equation converges to the solution of the kdv equation when the rotation parameter goes to 0 and the initial data of the kdv equation is in l2 to show this result we prove a bilinear estimate which is uniform with respect to

oleinik type estimates for the ostrovsky hunter equation abstract the ostrovsky hunter equation provides a model of small amplitude long waves in a rotating fluid of finite depth this is a nonlinear evolution equation in this study we consider the well posedness of the cauchy problem associated with this equation within a class of bounded discontinuous solutions we show that we can replace the kruzkov type entropy inequalities with an oleinik type estimate and we prove the uniqueness via a nonlocal adjoint problem this implies that a shock wave in an entropy weak solution to the ostrovsky hunter equation is admissible only if it jumps down in value similar to the inviscid burgers equation

radical rings with engel conditions abstract an associative ring r without unity is called radical if it coincides with its jacobson radical which means that the set of all elements of r forms a group denoted by r under the circle operation r s r s rs on r it is proved that for a radical ring r the group r satisfies an n engel condition for some positive integer n if and only if r is m engel as a lie ring for some positive integer m depending only on n

on the homology of associative algebras we present a new free resolution for k as an g module where g is an associative augmented algebra over a field k the resolution reflects the combinatorial properties of g introduction let k be a field and let g be an associative augmented k algebra for many purposes one wishes to have a projective resolution of k as a g module the bar resolution is always easy to define but it is often too large to use in practice at the other extreme minimal resolutions may exist but they are often hard to write down in a way that is amenable to calculations the main theorem of this paper presents a compromise resolution though rarely minimal it is small enough to offer some bounds but explicit enough to facilitate calculations as it relies heavily upon combinatorial constructions it is best suited for analyzing otherwise tricky algebras given via generators and relations since several results we get as consequences of the main theorem have been obtained before through other means this paper may be viewed as generalizing and unifying several seemingly unrelated ideas in particular we are generalizing priddy s results on koszul algebras 12 extending homology computations by govorov 9 and backelin 3 and complementing bergman s methods regarding the diamond lemma 6 three results may be of interest the homology of the mod p steenrod algebra is given in terms of the homology of a new chain complex smaller than the a algebra in theorem 3 5 formula 16 offers an efficient algorithm for the determination of hilbert series and theorem 4 2 asserts the existence of new bounds on the torsion groups of commutative graded rings 1 definitions and the main theorem throughout this paper k denotes any field and g is an associative k algebra with unity the field k embeds in g via 77 k g and we suppose that g has an augmentation i e a k algebra map s g k for which 77 is a right inverse s denotes a set of generators for g as a k algebra and k s is the free associative k algebra with unity on s there is a canonical surjection f k s g once s is chosen and the augmentation e is determined once we know s x for each x e s in particular this means that k s may be augmented such that f becomes a map of augmented algebras to s we associate a function e s z called a grading in the absence of a more compelling choice we often take e to be grading by length i e e x 1 for received by the editors may 23 1983 and in revised form february 22 1984 this paper was the subject of an invited one hour address in boulder colorado during the week long ams summer program on combinatorics and algebra june 1983 1980 mathematics subject classification 1985 revision primary 16a62 secondary 13d03 55s10 1986 american mathematical society 0002 9947 86 1 00 25 per page

well posedness of the cauchy problem of ostrovsky equation in anisotropic sobolev spaces abstract we study the cauchy problem of the ostrovsky equation t u x 3 u x 1 u u x u 0 with 0 by establishing a bilinear estimate on the anisotropic bourgain space x s b we prove that the cauchy problem of this equation is locally well posed in the anisotropic sobolev space h s r for any s 5 8 and some 0 1 2 using this result and conservation laws of this equation we also prove that the cauchy problem of this equation is globally well posed in h s r for s 0

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

quadratic approximation of solutions for boundary value problems with nonlocal boundary conditions in this paper using the quasilinearization method coupled with the method of upper and lower solutions we study a class of second order nonlinear boundary value problems with nonlocal boundary conditions we establish some sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem an example is also included to illustrate the main result

pi algebras generated by nilpotent elements of bounded index abstract let r be an associative algebra over a field f of positive characteristic p we address the following problem if r is generated by nilpotent elements with bounded index under what conditions can we conclude that r itself is nil of bounded index we prove that whenever r satisfies the engel condition and p 0 or r is lie soluble and p 2 then the conclusion holds along the way we prove that if r satisfies the engel condition then there exists a positive integer n such that the map x x p n is additive the converse also holds when f is infinite or r is generated by nilpotent elements

cauchy problem for the ostrovsky equation in spaces of low regularity abstract in this article we consider the initial value problem for the ostrovsky equation t u x 3 u x 1 u u x u 0 x r t r u x 0 u 0 x with initial data in sobolev spaces h s r using bourgain spaces we prove that the problem is locally well posed for s 1 2 if the sign of the third term of the equation is and for s 3 4 if the sign of this term is

continuity properties of the solution map for the generalized reduced ostrovsky equation abstract it is shown that the data to solution map for the generalized reduced ostrovsky gro equation is not uniformly continuous on bounded sets in sobolev spaces on the circle with exponent s 3 2 considering that for this range of exponents the gro equation is well posed with continuous dependence on initial data this result makes the continuity of the solution map an optimal property however if a weaker h r topology is used then it is shown that the solution map becomes holder continuous in h s

structure of prime finitely presented monomial algebras abstract the structure of a finitely presented monomial algebra k x k i over a field k is described here x is a finitely generated free monoid and i is a prime ideal of x that is finitely generated as an application a new structural proof of the recent result of bell and pekcagliyan j bell p pekcagliyan primitivity of finitely presented monomial algebras preprint arxiv 0712 0815v1 on the primitivity of such algebras is presented which yields a positive solution to the trichotomy problem raised by bell and smoktunowicz j bell a smoktunowicz the prime spectrum of algebras of quadratic growth j algebra 319 2008 414 431 in the finitely presented case our approach is based on a new result on the form of prime rees factors of semigroups satisfying the ascending chain condition on one sided annihilators and on its refinement in the case of finitely presented factors of the form x i

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

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation