on the descending central sequence of absolute galois groups let p be an odd prime number and f a field containing a primitive p th root of unity we prove a new restriction on the group theoretic structure of the absolute galois group g f of f namely the third subgroup g f 3 in the descending p central sequence of g f is the intersection of all open normal subgroups n such that g f n is 1 bbb z p 2 or the extra special group m p 3 of order p 3 and exponent p 2

massey products and ideal class groups we consider certain massey products in the cohomology of a galois extension of fields with coefficients in p power roots of unity we prove formulas for these products both in general and in the special case that the galois extension in question is the maximal extension of a number field unramified outside a set of primes s including those above p and any archimedean places we then consider those zp kummer extensions l of the maximal p cyclotomic extension k of a number field k that are unramified outside s we show that massey products describe the structure of a certain decomposition free quotient of a graded piece of the maximal unramified abelian pro p extension of l in which all primes above those in s split completely with the grading arising from the augmentation filtration on the group ring of the galois group of l k we explicitly describe examples of the maximal unramified abelian pro p extensions of unramified outside p kummer extensions of the cyclotomic field of all p power roots of unity for irregular primes p

critical point theory for nonsmooth functionals abstract in this paper we develop critical point theory for nonsmooth functional f h 0 1 r defined by f u 1 2 i j 1 a i j x u d i u d j u d x g x u d x the corresponding deformation lemmas are proved with the application to the bifurcation for quasilinear schrodinger equations in mind we extend the obtained results to that for a functional defined on a product space and prove a generalized saddle point theorem

quotients of absolute galois groups which determine the entire galois cohomology for a prime power q pd and a field f containing a root of unity of order q we show that the galois cohomology ring h g f mathbb z q is determined by a quotient g f 3 of the absolute galois group gf related to its descending q central sequence conversely we show that g f 3 is determined by the lower cohomology of gf this is used to give new examples of pro p groups which do not occur as absolute galois groups of fields

abelian 2 class field towers over the cyclotomic mathbb z _2 extensions of imaginary quadratic fields for the cyclotomic mathbb z 2 extension k of an imaginary quadratic field k we consider whether the galois group g k of the maximal unramified pro 2 extension over k is abelian or not the group g k is abelian if and only if the nth layer of the mathbb z 2 extension has abelian 2 class field tower for all n 1 the purpose of this paper is to classify all such imaginary quadratic fields k in part by using iwasawa polynomials

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

higher triple massey products and symbols abstract fix an arbitrary prime p let f be a field containing a primitive p th root of unity with absolute galois group g f and let h n denote its mod p cohomology group h n g f z p z the triple massey product of weight n k m n 3 is a partially defined multi valued function h n h k h m h n k m 1 in this work we prove that for an arbitrary prime p any defined 3mp of weight n 1 m where the first and third entries are symbols contains zero and that any defined 3mp of weight 1 k 1 where the middle entry is a symbol contains zero

asymptotic stability for a linear difference equation with variable delay we shall obtain sufficient conditions for the uniform stability and the global asymptotic stability of the linear difference equation with variable delay xn 1 xn pnxn kn 0 n n where pn is a sequence of nonnegative real numbers kn is a sequence of nonnegative integers and there exists a nonnegative integer k such that kn k for n n

triple massey products of weight 1 n 1 in galois cohomology abstract fix an arbitrary prime p let f be a field containing a primitive p th root of unity with absolute galois group g f and let h n denote its mod p cohomology group h n g f z p z the triple massey product abbreviated 3mp of weight n k m n 3 is a partially defined multi valued function h n h k h m h n k m 1 in this work we prove that for an odd prime p any defined 3mp of weight 1 k 1 contains zero

periodic solutions of difference equations abstract consider the difference equation x n 1 x n g x n n 0 1 arising as a discrete time network of single neuron where is the internal decay rate g is a signal function with mcculloch pitts nonlinearity for any positive integers k and m necessary and sufficient conditions are obtained for has a periodic solution of period 2 k 1 or 2 m in addition sufficient conditions are also given for has a 2 k 1 2 m periodic solution

multiple solutions for a quasilinear schrodinger equation abstract in this paper we consider the quasilinear schrodinger equation u v x u u 2 u g x u x r n where g and v are periodic in x 1 x n and g is odd in u subcritical and satisfies a monotonicity condition we employ the approach developed in szulkin and weth 2009 2010 15 16 and obtain infinitely many geometrically distinct solutions

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

periodic orbits on discrete dynamical systems abstract in this paper we discuss the discrete dynamical system x n 1 x n g x n n 0 1 arising as a discrete time network of single neuron where is the internal decay rate g is a signal function first we consider the case where g is of mcculloch pitts nonlinearity periodic orbits are discussed according to different range of moreover we can construct periodic orbits then we consider the case where g is a sigmoid function sufficient conditions are obtained for has periodic orbits of arbitrary periods and an example is also given to illustrate the theorem

convergence and periodicity in a discrete time network of two neurons with self connections abstract in this paper we study a class of difference systems modelling a two neuron network with feedback and mcculloch pitts type activation which includes the discrete version of the two neuron model with piecewise constant argument as its special case we obtain some interesting sufficient conditions insuring the convergence of solutions as well as the existence of periodic solutions

bifurcations for quasilinear schrodinger equations i abstract this paper is concerned with bifurcation problems for quasilinear schrodinger type equations such as the modified nonlinear schrodinger equations which have appeared as several models from mathematical physics we give a new approach for dealing with bifurcation problems of nonlinear potential operators the novelty of our approach lies in establishing multiple bifurcations without making a reduction to finite dimensional problems the finite dimensional reduction procedures that have been used in the past work for semilinear problems but are not suitable for quasilinear problems we give a unified approach requiring much less regularity of the problems as applications we provide multiple bifurcation results for the modified nonlinear schrodinger equations

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

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

asymptotic behavior of delay difference systems this paper is concerned with a delay difference system where k is a positive integer and f is a signal transmission function of mcculloch pitts type the difference system can be regarded as the discrete analog of the artificial neural network of two neurons with mcculloch pitts nonlinearity some interesting results are obtained for the asymptotic behavior of the system

all solutions of a class of discrete time systems are eventually periodic this paper is concerned with a class of discrete time systems as a model for two neurons with no internal decay such a discrete time system can be regarded as the discrete analog of a differential equation with piecewise constant nonlinearity of mcculloch pitts type base on some simple geometric facts we show that all solutions of this discrete time system are eventually periodic

critical points for multiple integrals of the calculus of variations in this paper we deal with the existence of critical points of functional defined on the sobolev space w01 p p 1 by j u int limits omega vartheta x u du dx text where is a bounded open subset of n even for very simple examples in n the differentiability of j u can fail to overcome this difficulty we prove a suitable version of the ambrosetti rabinowitz mountain pass theorem applicable to functionals which are not differentiable in all directions existence and multiplicity of nonnegative critical points are also studied through the use of this theorem

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

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

construction of unipotent galois extensions and massey products abstract for all primes p and for all fields we find a sufficient and necessary condition of the existence of a unipotent galois extension of degree p 6 the main goal of this paper is to describe an explicit construction of such a galois extension over fields admitting such a galois extension this construction is surprising in its simplicity and generality the problem of finding such a construction has been left open since 2003 recently a possible solution of this problem gained urgency because of an effort to extend new advances in galois theory and its relations with massey products in galois cohomology

infinitely many radial and non radial solutions to a quasilinear schrodinger equation abstract in this paper we are concerned with the following quasilinear schrodinger equation u u u 2 v x u f x u x r n by using a change of variables we obtained the existence of a sequence of radial solutions for n 2 a sequence of nonradial solutions for n 4 or n 6 and a nonradial solution for n 5

on tame pro p galois groups over basic mathbb z _p extensions for a prime number p and a finite set s of prime numbers congruent to 1 modulo p we consider the galois group of the maximal pro p extension unramified outside s over the mathbb z p extension of the rational number field in this paper we give a family of s for which the galois group is a metacyclic pro p group with an application to greenberg s conjecture

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

description of galois unipotent extensions abstract given an arbitrary field f we describe all galois extensions l f whose galois groups are isomorphic to the group of upper triangular unipotent 4 by 4 matrices with entries in the field of two elements

global attractivity of neutral difference equations abstract in this paper we discuss the global attractivity of the neutral difference equation n c n k p n n l 0 n 0 1 2 where is the forward difference operator c 1 1 k l and pn is a sequence of nonnegative real numbers

uniform stability of nonlinear difference systems abstract we offer sufficient conditions for the uniform stability and uniformly asymptotic stability of the zero solution of the nonlinear difference system formula where 0 1 f n s r sis the set of all functions n k 0 rfor somek n andxn sis defined byxn m x n m form n k 0

the kummerian property and maximal pro p galois groups abstract for a prime number p we give a new restriction on pro p groups g which are realizable as the maximal pro p galois group g f p for a field f containing a root of unity of order p this restriction arises from kummer theory and the structure of the maximal p radical extension of f we study it in the abstract context of pro p groups g with a continuous homomorphism g 1 p z p and characterize it cohomologically and in terms of 1 cocycles on g this is used to produce new examples of pro p groups which do not occur as maximal pro p galois groups of fields as above

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

stable periodic orbits in nonlinear discrete time neural networks with delayed feedback we consider a nonlinear discrete time system x n 1 x n g y n k y n 1 y n g x n k n n arising as a discrete time network of two neurons with mcculloch pitts nonlinearity where 0 1 k is a positive integer and g is a signal transmission function with a threshold we obtain a stable 4 k 1 periodic orbit in some regions of the parameters and we describe asymptotic behaviors of the system in other regions of

the galois action on geometric lattices and the mod ell i om this paper studies the galois action on a special lattice of geometric origin which is related to mod ell abelian by central quotients of geometric fundamental groups of varieties as a consequence we formulate and prove the mod ell abelian by central variant strengthening of a conjecture due to ihara oda matsumoto

multibump solutions for quasilinear elliptic equations abstract the current paper is concerned with constructing multibump type solutions for a class of quasilinear schrodinger type equations including the modified nonlinear schrodinger equations our results extend the existence results on multibump type solutions in coti zelati and rabinowitz 1992 17 to the quasilinear case our work provides a theoretic framework for dealing with quasilinear problems which lack both smoothness and compactness by using more refined variational techniques such as gluing techniques morse theory lyapunov schmidt reduction etc

multibump solutions and critical groups we consider the newtonian system ddot q b t q w q q t with b w periodic in t b positive definite and show that for each isolated homoclinic solution q 0 having a nontrivial critical group in the sense of morse theory multibump solutions with 2 le k le iy bumps can be constructed by gluing translates of q 0 further we show that the collection of multibumps is semiconjugate to the bernoulli shift next we consider the schr odinger equation delta u v x u g x u in rn where v g are periodic in x 1 ldots x n sigma delta v subset 0 iy and we show that similar results hold in this case as well in particular if g x u u 2 2 u n ge 4 and v changes sign then there exists a solution minimizing the associated functional on the nehari manifold this solution gives rise to multibumps if it is isolated

multibump solutions for quasilinear elliptic equations with critical growth the current paper is concerned with constructing multibump solutions for a class of quasilinear schrodinger equations with critical growth this extends the classical results of coti zelati and rabinowitz commun pure appl math 45 1217 1269 1992 for semilinear equations as well as recent work of liu wang and guo j funct anal 262 4040 4102 2012 for quasilinear problems with subcritical growth the periodicity of the potentials is used to glue ground state solutions to construct multibump bound state solutions

massey products for elliptic curves of rank 1 for an elliptic curve over q of analytic rank 1 we use the level two selmer variety and secondary cohomology products to find explicit analytic defining equations for global integral points inside the set of p adic points

periodic orbits of a neuron model with periodic internal decay rate in this paper we will study a non autonomous piecewise linear difference equation which describes a discrete version of a single neuron model with a periodic internal decay rate we will investigate the periodic behavior of solutions relative to the periodic internal decay rate furthermore we will show that only periodic orbits of even periods can exist and show their stability character

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

solutions for a quasilinear schrodinger equation a dual approach abstract we consider quasilinear stationary schrodinger equations of the form 1 u u 2 u g x u x r n introducing a change of unknown we transform the search of solutions u x of 1 into the search of solutions v x of the semilinear equation 2 v 1 1 2f 2 v g x f v x r n where f is suitably chosen if v is a classical solution of 2 then u f v is a classical solution of 1 variational methods are then used to obtain various existence results

dynamics of a class of nonlinear discrete time neural networks in this paper we consider a class of delay difference systems with piecewise constant nonlinearity which includes the discrete version of an artificial neural network model of two neurons described by differential equations with piecewise constant arguments some interesting results are obtained for the periodicity and convergence of solutions of the systems

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

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