conical zeta values and their double subdivision relations abstract we introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones the quasi shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values in order to achieve the closed cone subdivision relation we also interpret linear relations among fractions as subdivisions of decorated closed cones as a generalization of the double shuffle relation of multiple zeta values we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values

the determination of the lie algebra associated to the lower central series of a group in this paper we determine the lie algebra associated to the lower central series of a finitely presented group in the case where the defining relators satisfy certain independence conditions other central series such as the lower p central series are treated as well 1 statement of results the main purpose of this paper is to determine the lie algebra associated to the lower central series of a group thus extending the results of 6 7 to groups defined by more than one relator the methods apply to other central series such as the tower p central series the lower central series of a group g is the sequence of subgroups g 1 n 1 defined inductively by g g gi1 g g where g gn denotes the subgroup of g generated by the commutators x y xy 1xy with x e g y c gn the associated graded abelian group gr g dn 1grn g where grn g gn gn 1 has the structure of a graded lie algebra over the ring z of integers the bracket operation in gr g being induced by the commutator operation in g cf 2 9 11 12 the construction of the lie algebra gr g uses only the fact that gn is a sequence of subgroups of g with the following properties i g1 g ii gn 1 c gn iii gn9gkl c gn k such a family of subgroups of g is called a central filtration of g let f be the free group on the n letters x19 xn and let f be the lower central series of f if i is the image of xi in gr1 f f f f then the lie algebra received by the editors may 10 1983 invited paper presented at the special session in combinatorial group theory at the 803rd meeting of the ams in new york city april 14 15 1983 1980 mathematics subject classification primary 20f14 20f40 secondary 20e18

a comodule bialgebra structure for word series substitution and mould composition abstract an internal coproduct is described which is compatible with hoffman s quasi shuffle product hoffman s quasi shuffle hopf algebra with deconcatenation coproduct is a comodule hopf algebra over the bialgebra thus defined the relation with ecalle s mould calculus i e mould composition and contracting arborification is precised

the algebra of multiple harmonic series abstract recently there has been much interest in multiple harmonic series i 1 i 2 i k n 1 n 2 n k 1 1 n 1 i 1 n 2 i 2 n k i k which converge when the exponentsijare positive integers andi1 1 also known as multiple zeta values or euler zagier sums starting with the noncommutative polynomial algebraq x y we define a second multiplication which is commutative and associative and call the resulting structure the harmonic algebra h as a graded commutative algebra h turns out to be a free polynomial algebra with the number of generators in degreengiven by the witt formula for the numbern n of basis elements of degreenin the free lie algebra on two generators multiple harmonic series can be thought of as images under a map h 0 rwhich is a homomorphism with respect to the commutative multiplication where h 0is an appropriate subalgebra of h if we calli1 ikthe weight of the series i1 ik then forn 1 there are at mostn n series of weightnthat are irreducible in the sense that they are not sums of rational multiples of products of multiple harmonic series of lower weight in fact our approach gives an explicit set of algebraically irreducible multiple harmonic series we also show that there is a subalgebra h 0 h 1 h related to the shuffle algebra which contains the algebra s of symmetric functions in fact maps the elements of s h 0to algebraic combinations of zeta values i for integeri 2 the map is not injective we show how several results about multiple harmonic series can be recast as statements about the kernel of and propose some conjectures on the structure of the algebra h 0 ker

sur la structure galoisienne de certaines pro p extensions de corps de nombres soit k k la pro p extension maximale du corps de nombres k non ramifiee en dehors d un ensemble fini de places de k soit mathcal h un sous groupe ferme et distingue de g gal k k et soit rm g rm g sigma mathcal h le quotient dans des contextes arithmetiques dependant de et de la structure de g nous etudions ici la liberte du mathbb z p g module mathcal x mathcal h mathcal h mathcal h

the zassenhaus filtration massey products and representations of profinite groups abstract we consider the p zassenhaus filtration g n of a profinite group g suppose that g s n for a free profinite group s and a normal subgroup n of s contained in s n under a cohomological assumption on the n fold massey products which holds e g if g has p cohomological dimension 1 we prove that g n 1 is the intersection of all kernels of upper triangular unipotent n 1 dimensional representations of g over f p this extends earlier results by minac spira and the author on the structure of absolute galois groups of fields

golod shafarevich algebras free subalgebras and noetherian images abstract it is shown that golod shafarevich algebras of a reduced number of defining relations contain noncommutative free subalgebras in two generators and that these algebras can be homomorphically mapped onto prime noetherian algebras with linear growth it is also shown that golod shafarevich algebras of a reduced number of relations cannot be nil

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

cohomologically generic 2 complexes and 3 dimensional poincare complexes this is part of a series of papers devoted to topological applications of deformation theory imp mp2 especially in low dimensional topology bp mp1 our lowdimensional program consists in fixing accessible homological invariants like the betti numbers or the cup product structure of the cohomology and then apply the machinery of deformation theory in order to both compute and distinguish more interesting invariants such as the graded lie algebra associated to the fundamental group gr l itself the entire homotopy type here we address the problem of computing gr ttl s a difficult task in general in terms of h s z the forthcoming bp and imp1 deal with the other above mentioned invariants leading in particular to invariants associated to links and 3 manifolds via deformation theory one may associate to any connected space s assuming h s z to be free of finite rank a graded lie algebra l defined as the quotient of the free lie algebra on has by the ideal generated by the image of the dual of the cup product h sah s h2s see sect 1 our main result reads

solitonic excitations and interactions in an α helical protein modeled by three coupled nonlinear schrodinger equations with variable coefficients governing the molecular excitations associated with bioenergy transport in an helical protein through the soliton modes three coupled higher order nonlinear schrodinger equations with variable coefficients are investigated via symbolic computation using bell polynomials a bilinear form and backlund transformation are derived for this model furthermore explicit n soliton solutions are constructed in terms of the double wronskian determinant solitonic excitations are found to remain stable against the disorders of the parameters under certain constraints finally propagation characteristics and interactions of the solitonic excitations are discussed soliton amplitudes are related to the energy modulation coefficients and both the soliton properties and energy distribution during the interactions are affected by the inhomogeneity coefficients of the protein with time dependent disorders

mild pro p groups with 4 generators abstract let p be an odd prime and s a finite set of primes 1 mod p we give an effective criterion for determining when the galois group g g s p of the maximal p extension of q unramified outside of s is mild when s 4 and the cup product h 1 g z p z h 1 g z p z h 2 g z p z is surjective

non commutative graded algebras and their hilbert series abstract this paper begins to develop a theory of non commutative graded algebras and their hilbert series which parallels the extensive already existing theory on commutative graded algebras we begin with a discussion of order ideals of monomials which give us vector space bases for these algebras we develop next the concept of weak summand which is a special relationship a subalgebra may have to an algebra and the concept of strongly free sets strongly free sets have many properties which are analogous to the properties of regular sequences in commutative algebras we discuss combinatorially free sets which are useful for constructing examples of many of the preceding ideas we conclude with an application of our results to algebraic topology by showing that there is a finite cw complex with only nine positive dimensional cells whose loop space has an irrational poincare series

wronskian and grammian solutions for a 2 1 dimensional date jimbo kashiwara miwa equation abstract in this paper a 2 1 dimensional date jimbo kashiwara miwa equation is investigated based on the hirota method and auxiliary variable backlund transformation is obtained under certain conditions the hirota bilinear forms can be reduced to the plucker and jacobi identities via the wronskian and grammian solutions through the wronskian technique and pfaffian derivative formulae n soliton solutions in the wronskian and grammian are given graphically presented soliton collisions are elastic both in the wronskian and grammian and after each collision the solitons keep their amplitudes unchanged except for the phase shifts

mild pro 2 groups and 2 extensions of q with restricted ramification abstract using the mixed lie algebras of lazard we extend the results of the first author on mild groups to the case p 2 in particular we show that for any finite set s 0 of odd rational primes we can find a finite set s of odd rational primes containing s 0 such that the galois group of the maximal 2 extension of q unramified outside s is mild we thus produce a projective system of such galois groups which converge to the maximal pro 2 quotient of the absolute galois group of q unramified at 2 and our results also allow results of alexander schmidt on pro p fundamental groups of marked arithmetic curves to be extended to the case p 2 over a global field which is either a function field of characteristic 2 or a totally imaginary number field

bell polynomial manipulations on the backlund transformations and lax pairs for some soliton equations with one tau function in the framework of bell polynomial manipulations under investigation hereby are three single field bilinearizable equations the 1 1 dimensional shallow water wave model boiti leon manna pempinelli model and 2 1 dimensional sawada kotera model based on the concept of scale invariance a direct and unifying bell polynomial scheme is employed to achieve the backlund transformations and lax pairs associated with those three soliton equations note that the bell polynomial expressions and bell polynomial typed backlund transformations for those three soliton equations can be respectively cast into the bilinear equations and bilinear backlund transformations with symbolic computation consequently it is also shown that the bell polynomial typed backlund transformations can be linearized into the corresponding lax pairs

growth rates _ homology and volumes of hyperbolic 3 manifolds it is shown that if m is a closed orientable irreducible 3 manifold and n is a nonnegative integer and if h 1 m z p has rank n 2 for some prime p then every n generator subgroup of 1 m has infinite index in 1 m and is in fact contained in infinitely many finite index subgroups of 1 m this result is used to estimate the growth rates of the fundamental group of a 3 manifold in terms of the rank of the z p homology

bell polynomial approach applied to the seventh order sawada kotera ito equation in this paper the bell polynomial approach is extended to the seventh order sawada kotera ito equation which is regarded as a higher order integrable one in the korteweg de vries hierarchy through several combinations of the constructed bell polynomials the bilinear form with an auxiliary independent variable introduced backlund transformation lax pair and infinite conservation laws for such an equation are derived by means of symbolic computation

soliton and antisoliton interactions in the good boussinesq equation the solitary wave interaction mechanism for the good boussinesq equation is investigated and found to be far more complicated than was previously thought three salient features are that solitary waves only exist for a finite range of velocities that large solitons can turn into so called antisolitons and that it is possible for solitons to merge and split small solitons however appear to be stable the existence of a potential well is linked to the different behaviors observed between small and large initial conditions

pure virtual braids resonance and formality we investigate the resonance varieties lower central series ranks and chen ranks of the pure virtual braid groups and their upper triangular subgroups as an application we give a complete answer to the 1 formality question for this class of groups in the process we explore various connections between the alexander type invariants of a finitely generated group and several of the graded lie algebras associated to it and discuss possible extensions of the resonance chen ranks formula in this context

periodic free resolutions from twisted matrix factorizations abstract the notion of a matrix factorization was introduced by eisenbud in the commutative case in his study of bounded periodic free resolutions over complete intersections since then matrix factorizations have appeared in a number of applications in this work we extend the notion of homogeneous matrix factorizations to regular normal elements of connected graded algebras over a field next we relate the category of twisted matrix factorizations to an element over a ring and certain zhang twists we also show that in the setting of a quotient of a ring of finite global dimension by a normal regular element every sufficiently high syzygy module is the cokernel of some twisted matrix factorization furthermore we show that in the noetherian as regular setting there is an equivalence of categories between the homotopy category of twisted matrix factorizations and the singularity category of the hypersurface following work of orlov

optimal 5 step nilpotent quadratic algebras by the golod shafarevich theorem an associative algebra r given by n generators and d n23 homogeneous quadratic relations is not 5 step nilpotent we prove that this estimate is optimal namely we show that for every positive integer n there is an algebra r given by n generators and n23 homogeneous quadratic relations such that r is 5 step nilpotent

on the combinatorics of several integrable hierarchies we demonstrate that statistics of certain classes of set partitions are described by generating functions related to the burgers ibragimov shabat and korteweg de vries integrable hierarchies

improved bell polynomial procedure for the higher order korteweg de vries equations in fluid dynamics korteweg de vries kdv typed equations are seen in fluid dynamics plasma physics and other fields by means of the bell polynomial procedure we take two higher order members of the kdv hierarchy the seventh and ninth order lax s kdv equations in fluid dynamics as the examples for studying the integrable properties of the higher order equations different from lower order equations two new partial differential operators in the bell polynomial procedure are introduced to construct the multi dimensional bilinear forms of such equations with several auxiliary independent variables through the procedure simplified via the algebraic operation of the polynomials the backlund transformations lax pairs and infinite conservation laws of such equations are deduced

group presentation p adic analytic groups and lattices in mathrm sl _2 mathbf c groups by using their pro p completions this method is based on the observation that if x r is a presentation of the finitely generated f g group f then x r considered as a presentation for a pro p group is actually a presentation for t the pro p completion of f first we prove a generalization of a theorem of golod and shafarevich cf h theorem let g be a p adic analytic pro p group different from zp the p adic integers if x r is a minimal presentation of g then 1x 2 iri 2

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

trees and discrete subgroups of lie groups over local fields let k be a locally compact field and g a simple at group g g k a discrete subgroup t of g is called a lattice if g f carries a finite g invariant measure it is a uniform or cocompact lattice if g t is compact and nonuniform otherwise when the jrf rank of g is greater than one margulis ma z proved that t is arithmetic establishing the conjecture of selberg and piatetskishapiro this remarkable work left open the case of rank one groups sli u contains continuous families of lattices the teichmuller spaces and in particular it contains nonarithmetic lattices by the mostow rigidity theorem cf ml sli r is essentially the only real simple lie group which allows this phenomenon some other real rank one groups are known to have nonarithmetic lattices gromov and piatetski shapiro gps showed that so n 1 have such lattices earlier it was shown by makarov and vinberg for small n mostow m2 constructed nonarithmetic lattices in su 2 1 and together with deligne dm also in su 3 1 for the other rank one real groups su n 1 n 4 sp n 1 and f4 the problem is still open a related problem is the congruence subgroup problem which asked given t arithmetic are all its finite index subgroups congruence subgroups serre si conjectured that t has the congruence subgroup property csp if and only if rank g 2 the affirmative part of this conjecture was proved to a large extent but mainly for nonuniform lattices see ri r2 for precise results and history less is known for arithmetic lattices in real rank one groups it is easy to prove that none of the arithmetic lattices in psl2 r so 2 1 has csp the same holds for psl2 c so 3 1 see si for the nonuniform case and li for the uniform case for so n 1 general n it is known only for some of the lattices millson mi similarly kazdhan ka showed that some lattices in su n 1 do not have csp again nothing is known for sp n 1 and which are of rank one but have kazdhan property t like groups of higher rank in this note we will describe results on the structure of lattices in rank one groups over locally compact nonarchimedean fields as well as some

mild pro p groups and galois groups of p extensions of ℚ in this paper we introduce a new class of finitely presented pro pgroups g of cohomological dimension 2 called mild groups if d g r g are respectively the minimal number of generators and relations of g we give an infinite family of mild groups g with r g d g and d g 2 arbitrary these groups can be constructed with g g g finite answering a question of kuzmin if g gs p is the galois group of the maximal p extension of q unramified outside a finite set of primes s and p 6 2 we show that g is mild for a co final class of sets s even in the case p s

the notion of dimension in geometry and algebra this talk reviews some mathematical and physical ideas related to the notion of dimension after a brief historical introduction various modern constructions from fractal geometry noncommutative geometry and theoretical physics are invoked and compared glenn gould disapproved of his own recording of goldberg variations there is a lot of piano playing going on there and i mean that as the most disparaging comment possible nyrb oct 7 2004 p 10

higher massey products in the cohomology of mild pro p groups abstract translating results due to j labute into group cohomological language a schmidt proved that a finitely presented pro p group g is mild and hence of cohomological dimension cd g 2 if h 1 g f p u v as f p vector space and the cup product h 1 g f p h 1 g f p h 2 g f p maps u v surjectively onto h 2 g f p and is identically zero on v v this has led to important results in the study of p extensions of number fields with restricted ramification in particular in the case of tame ramification in this paper we extend labute s theory of mild pro p groups with respect to weighted zassenhaus filtrations and prove a generalization of the above result for higher massey products which allows to construct mild pro p groups with defining relations of arbitrary degree we apply these results to one relator pro p groups and obtain some new evidence of an open question due to serre

the entropy of graded algebras we define the entropy of a graded algebra a n an by limsupn diman this is related to the notion of entropy in symbolic dynamics and could serve as a natural dimension concept for algebras with exponential growth we study the entropy of quotients and subalgebras of free associative algebras and free lie algebras we also study the behavior of the entropy function under free products and obtain several characterizations of free algebras in terms of entropy

bell polynomial approach and n soliton solution for the extended lotka volterra equation in plasmas symbolically investigated in this paper is the extended lotka volterra elv equation which can govern the kinetics of the discrete peaks of the weak langmuir turbulence in plasmas without the linear damping and random noise binary bell polynomials are applied to the bilinearization of the discrete system bilinear backlund transformation of the elv equation is constructed n soliton solution in terms of the extended casorati determinant is also presented and verified propagation and interaction behaviors of the langmuir turbulence are analyzed it is demonstrated that the number of the interacting langmuir waves can influence the soliton velocity and amplitude as well as the collision phase shift graphic illustrations of the solitonic collisions show that the repulsion effects and nonlinear interactions are also associated with the number of the interacting langmuir waves

lax pair backlund transformation and multi soliton solutions for the boussinesq burgers equations from shallow water waves abstract under investigation in this paper is the set of the boussinesq burgers bb equations which can be used to describe the propagation of shallow water waves based on the binary bell polynomials hirota method and symbolic computation the bilinear form and soliton solutions for the bb equations are derived backlund transformations bts in both the binary bell polynomial and bilinear forms are obtained through the bt in the binary bell polynomial form a type of solutions and lax pair for the bb equations are presented as well propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis shock wave and bell shape solitons are respectively obtained for the horizontal velocity field u and height v of the water surface in both the head on and overtaking collisions the shock waves for the u profile change their shapes which denotes that the collisions for the u profile are inelastic however the collisions for the v profile are proved to be elastic through the asymptotic analysis our results might have some potential applications for the harbor and coastal design

zeroth hochschild homology of preprojective algebras over the integers we determine the z module structure of the preprojective algebra and its zeroth hochschild homology for any non dynkin quiver and hence the structure working over any base commutative ring of any characteristic this answers and generalizes a conjecture of hesselholt and rains producing new p torsion classes in degrees 2pl l 1 we relate these classes by p th power maps and interpret them in terms of the kernel of verschiebung maps from noncommutative witt theory an important tool is a generalization of the diamond lemma to modules over commutative rings which we give in the appendix

double shuffle relations for arborified zeta values abstract arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees this approach allows to study their convergence domain and to relate them to multiple zeta values generalisations to rooted trees of the stuffle and shuffle products are defined and studied it is further shown that arborified zeta values are algebra morphisms for these new products on trees

on galois representations arising from towers of coverings of p1 0 1 we propose a new way to describe universally thel adic galois representations associated to each almost pro l tower of etale coverings ofp1 0 1 this generalizes our universal power series for jacobi sums cf i which arises from the tower of fermat curves of degreeln n and contains the case of the tower of modular curves of level 2mln m fixed n as another important special case as a fundamental tool we shall establish and use an almost pro l version of the theorems of blanchfield and of lyndon in fox free differential calculus

on monomial algebras of finite global dimension let g be an associative monomial k algebra if g is assumed to be finitely presented then either g contains a free subalgebra on two monomials or else g has polynomial growth if instead g is assumed to have finite global dimension then either g contains a free subalgebra or else g has a finite presentation and polynomial growth also a graded hopf algebra with generators in degree one and relations in degree two contains a free hopf subalgebra if the number of relations is small enough

loop space homology of spaces of small category only little is known concerning h omegax k the loop space homology of a finite cw complex x with coefficients in a field k a space x is called an r cone if there exists a filtration x0 subset of x1 subset of subset of x r x such that x i has the homotopy type of the cofibre of a map from a wedge of sphere into x i 1 denote by a x the sub hopf algebra image of h omegax1 we prove then that for a graded r cone r less than or equal to 3 there exists an isomorphism a x x t u congruent to h omegax

covering spaces of 3 orbifolds let o be a compact orientable 3 orbifold with non empty singular locus and a finite volume hyperbolic structure equivalently the interior of o is the quotient of hyperbolic 3 space by a lattice in psl 2 c with torsion then we prove that o has a tower of finite sheeted covers oi with linear growth of p homology for some prime p this means that the dimension of the first homology with mod p coefficients of the fundamental group of oi grows linearly in the covering degree the proof combines techniques from 3 manifold theory with group theoretic methods including the golodshafarevich inequality and results about p adic analytic pro p groups this has several consequences firstly the fundamental group of o has at least exponential subgroup growth secondly the covers oi have positive heegaard gradient thirdly we use it to show that a group theoretic conjecture of lubotzky and zelmanov would imply that o has large fundamental group this implication uses a new theorem of the author which will appear in a forthcoming paper these results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3 orbifold with non empty singular locus has large fundamental group many of the above results apply also to 3 manifolds commensurable with an orientable finite volume hyperbolic 3 orbifold with non empty singular locus this includes all closed orientable hyperbolic 3 manifolds with rank two fundamental group and all arithmetic 3 manifolds

graph products of spheres associative graded algebras and hilbert series given a finite simple vertex weighted graph we construct a graded associative noncommutative algebra whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges we show that the hilbert series of this algebra is the inverse of the clique polynomial of the graph using this result it easy to recognize if the ideal is inert from which strong results on the algebra follow noncommutative grobner bases play an important role in our proof there is an interesting application to toric topology this algebra arises naturally from a partial product of spheres which is a special case of a generalized moment angle complex we apply our result to the loop space homology of this space

on the integrable variant of the boussinesq system painleve property rational solutions a related many body system and equivalence with the akns hierarchy abstract an infinite family of rational solutions of the integrable version of the boussinesq system u t w x uu x 0 w t u xxx uw x 0 and the associated higher order flows is constructed the differential equations governing the motion of the poles are derived and their complete integrability demonstrated this system embeds into two disjoint calogero moser systems coupled only through a constraint a novel treatment of this constraint is given using the equivalence of to the real version of the nonlinear schrodinger system it is seen that rational solutions of the akns hierarchy have been constructed as well finally it is shown that all solutions of these hierarchies including the rational ones also solve the first modified kp equation

surface subgroups of kleinian groups with torsion we prove that every finitely generated kleinian group that contains a finite non cyclic subgroup either is finite or virtually free or contains a surface subgroup hence every arithmetic kleinian group contains a surface subgroup

inert sets and the lie algebra associated to a group abstract independently j labute trans amer math soc 288 1985 51 57 has given a sufficient condition for the lie algebra associated with a discrete group s lower central series to have a certain presentation and s halperin and j m lemaire math scand in press have discussed the conditions under which some homogeneous elements in a graded algebra constitute an inert set as it turns out these conditions essentially coincide as a consequence various theorems about graded algebras may now be translated into results into group theory a nontrivial application to link groups is offered

on a direct bilinearization method kaup s higher order water wave equation as a modified nonlocal boussinesq equation a systematic procedure for the bilinearization of classes of soliton equations is developed with the help of a generalization of bell s exponential polynomials 1934 application of this procedure to kaup s higher order wave equation 1975 discloses several links with other soliton systems in particular it is found that the kaup equation is the modified version of a sech square soliton system which constitutes an alternative to the good boussinesq equation

coherent algebras and noncommutative projective lines abstract a well known conjecture says that every one relator group is coherent we state and partly prove a similar statement for graded associative algebras in particular we show that every gorenstein algebra a of global dimension 2 is graded coherent this allows us to define a noncommutative analogue of the projective line p 1 as a noncommutative scheme based on the coherent noncommutative spectrum qgra of such an algebra a that is the category of coherent a modules modulo the torsion ones this category is always abelian ext finite hereditary with serre duality like the category of coherent sheaves on p 1 in this way we obtain a sequence p n 1 n 2 of pairwise non isomorphic noncommutative schemes which generalize the scheme p 1 p 2 1

kp constraints from reduced multi component hierarchies the m vector k constrained kadomtsev petviashvili kp hierarchy is shown to be a pseudo reduction of the m 1 component kp hierarchy to facilitate the implementation of this reduction on the level of the solutions the typical multi component kp solutions are mapped onto solutions of a toda molecule type equation from which wronskian and grammian solutions for the constrained kp hierarchy follow the reduction of the associated linear systems is discussed and its importance for the choice of bilinear representation of the reduced systems is explained

engulfing and subgroup separability for hyperbolic groups if a group is subgroup separable otherwise known as locally extended residually finite or lerf one can pass from immersions to embeddings in some finite covering space we show that a certain engulfing property gives subgroup separability for a large and useful class of subgroups of hyperbolic 3 manifold groups

free subalgebras of c algebras abstract the aim of this work is to study the existence of free subalgebras in c algebras the kurosh levitzky problem and related conjectures of makar limanov are answered in the context of c algebras in particular we characterize and study the existence of free non abelian subalgebras with two self adjoint contractions in finitely generated c algebras c algebras having linear growth are shown to be subhomogeneous

orientation reversing involutions and the first betti number for finite coverings of 3 manifolds letm s 3 p 3 be a closed orientable irreducible 3 manifold which admits an orientation reversing involution m m if dim fix 0 suppose 1 m has a subgroup of even index we show thatm has finite coverm mm with 1 m 0 as an application we show that the hyperbolic dodecahedral space has a finite cover with positive 1st betti number

integrability of the modified generalised vakhnenko equation integrability of the modified generalised vakhnenko equation is investigated systematically based on binary bell polynomials its bilinear representation n soliton solutions bilinear backlund transformation and lax pair are succinctly constructed moreover the conservation laws of the modified generalised vakhnenko equation are discussed by using corresponding lax pair furthermore the quasiperiodic solution of the modified generalised vakhnenko equation is presented by applying hirota direct method and riemann theta function the asymptotic behavior of the one periodic wave is analyzed in details it is shown that the one periodic wave solution tends to the one soliton solution under a small amplitude limit 0 finally the new n soliton solutions of the standard vakhnenko equation are presented it would be specially mentioned that all the results of modified generalised vakhnenko equation can be reduced to the generalised vakhnenko equation and standard vakhnenko equation under the special case of 1 and 1 0 respectively

bilinear backlund transformations kink periodic solitary wave and lump wave solutions of the bogoyavlenskii kadomtsev petviashvili equation abstract this paper investigates the bogoyavlenskii kadomtsev petviashvili equation by using the binary bell polynomials method and differential constraint technique several kinds of more general bilinear representations and baklund transformations are presented at the same time multiple wave solutions including the kink periodic solitary wave and bright dark lump wave solutions are constructed the resonant interactions between two kink solitary waves are investigated and exhibited mathematically and graphically lump wave solutions a kind of rational function localized in all directions of the space are obtained by the taylor expansion of the kink periodic solitary wave also we discuss the dynamical behavior of the lump wave solutions and their structures

circular sets of prime numbers and p extensions of the rationals zusammenfassung sei p eine ungerade primzahl und s eine endliche menge von primzahlen kongruent 1 modulo p wir zeigen da die gruppe gs q p von kohomologischer dimension 2 ist wenn das verschlingungsdiagramm zu s und p einer gewissen technischen bedingung genaugt und wir zeigen da gs q p in diesen faallen eine dualitaatsgruppe ist au erdem untersuchen wir das zerlegungsverhalten von primstellen in der erweiterung qs p q und wir setzen die kohomologie von gs q p zur etalkohomolgie des schemas spec z is in beziehung schlie lich berechnen wir den dualisierenden modul abstract let p be an odd prime number and let s be a nite set of prime numbers congruent to 1 modulo p we prove that the group gs q p has cohomological dimension 2 if the linking diagram attached to s and p satis es a certain technical condition and we show that gs q p is a duality group in these cases furthermore we investigate the decomposition behaviour of primes in the extension qs p q and we relate the cohomology of gs q p to the etale cohomology of the scheme spec z i s finally we calculate the dualizing module

a generalization of castelnuovo mumford regularity for representations of noncommutative algebras abstract we introduce and generalize the notion of castelnuovo mumford regularity for representations of noncommutative algebras effectively establishing a measure of complexity for such objects the grobner shirshov basis theory for modules over noncommutative algebras is developed by which a noncommutative analogue of schreyer s theorem is proved for computing syzygies by a repeated application of this theorem we construct free resolutions for representations of noncommutative algebras some interesting examples are included in which graded free resolutions and regularities are computed for representations of various algebras in particular using the bernstein gelfand gelfand resolutions for integrable highest weight modules over kac moody algebras we compute the projective dimensions and regularities explicitly for the cases of finite type and affine type a n 1

miura transformation between two non linear equations in 2 1 dimensions a dispersive wave equation in 2 1 dimensions 2ldw widely discussed by different authors is shown to be nothing but the modified version of the generalized dispersive wave equation gldw using singularity analysis and techniques based upon the painleve property leading to the double singular manifold expansion we shall find the miura transformation which converts the 2ldw equation into the gldw equation through this miura transformation we shall also present the lax pair of the 2ldw equation as well as some interesting reductions to several already known integrable systems in 1 1 dimensions as the 2ldw equation arises from a miura transformation we propose that it should be treated conventionally as a modified equation in this case we propose its designation as the mgldw equation

on hopf algebra structures over free operads the operad lie can be constructed as the operad of primitives primas from the operad as of associative algebras this is reflected by the theorems of friedrichs poincare birkhoff witt and cartier milnor moore we replace the operad as by families of free operads p which include the operad mag freely generated by a non commutative non associative binary operation and the operad of stasheff polytopes we obtain poincare birkhoff witt type theorems and collect information about the operads primp e g in terms of characteristic functions

bilinearization of the non local boussinesq equation a single field bilinear system generating the so called non local boussinesq equation is constructed from the bilinearization procedure it can be seen that the associated hierarchy of soliton systems which we construct shares part of the solution set of a hierarchy related to the kadomtsev petviashvili equation a bilinear representation of the recursion operator for the kaup hierarchy is essential in the construction and a systematic way of obtaining such a representation from just two soliton considerations is presented

from dynamics on surfaces to rational points on curves has no integer solutions with x y z 1 inspiring generations of work in number theory its proof was finally achieved by wiles a qualitative result finite fermat was obtained earlier by faltings it says the fermat equation has only a finite number of solutions for each given n up to rescaling this paper is an appreciation of some of the topological intuitions behind number theory it aims to trace a logical path from the classification of surface diffeomorphisms to the proof of finite fermat the route we take is the following

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

on modified nls kaup and nlbq equations differential transformations and bilinearization new transformations between the nonlinear schrodinger kaup and non local boussinesq equations as well as their modified counterparts are found and analysed the bilinear representations of these equations including an alternative bilinear form of the chen lee liu equation are obtained by a direct method based on the bell s exponential polynomials explicit wronskian solutions to these equations are also presented

ancestors graph and an upper bound for the subword complexity function a general upper bound for the subword complexity function of a fixed point of a primitive substitution is presented the approach relies on the introduction and discussion of the ancestors graph of a substitution more precisely we characterize the language of a fixed point of a primitive substitution as the strongly connected component of the letters in its ancestors graph we propose algorithms for the local construction of this graph we give an upper bound for the subword complexity function in terms of the hilbert series of some quotient of a noncommutative polynomial ring by a sum of automatic bilateral ideals

the kernel unipotent conjecture and the vanishing of massey products for odd rigid fields abstract a major difficult problem in galois theory is the characterization of profinite groups which are realizable as absolute galois groups of fields recently the kernel n unipotent conjecture and the vanishing n massey conjecture for n 3 were formulated these conjectures evolved in the last forty years as a byproduct of the application of topological methods to galois cohomology we show that both of these conjectures are true for odd rigid fields this is the first case of a significant family of fields where both of the conjectures are verified besides fields whose galois groups of p maximal extensions are free pro p groups we also prove the kernel unipotent conjecture for demushkin groups of rank 2 and establish various filtration results for free pro p groups provide examples of pro p groups which do not have the kernel n unipotent property compare various zassenhaus filtrations with the descending p central series and establish new type of automatic galois realization

dromion like structures in a 3 1 dimensional kdv type equation dromion structures have been found in many 2 1 dimensional models the similar structure in 3 1 dimensions is studied in this paper for a kdv type equation starting from the bilinear form of the model we found that there are five types of multi dromion solutions for its potentials say the first type of multi dromion solution is driven by multi camber solitons with one of them being non parallel to the x axis the second third and fourth types of multi dromion solutions are driven by two camber solitons and one two and three sets of parallel plane solitons respectively the fifth type of multi dromion solution is driven by one set of parallel plane solitons one set of camber solitons which are parallel to the x axis and one camber soliton which is non parallel to the x axis phase shifts may be involved in the interactions among the multi dromions for the first and fifth types of solutions a single dromion solution of the model may possess quite a free shape for instance the point like dromions ring type dromions extended and sharp dromions and oscillatory dromions can be obtained by selecting some arbitrary functions appropriately