sobolev type embedding and quasilinear elliptic equations with radial potentials abstract we study the existence of nontrivial radial solutions for quasilinear elliptic equations with unbounded or decaying radial potentials the existence results are based upon several new embedding theorems we establish in the paper for radially symmetric functions

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

an improved bonferroni inequality and applications summary we present an improved bonferroni inequality which gives an upper bound for the probability of the union of an arbitrary sequence of events the bound is constructed in terms of the joint probability of pairs of events which are represented by edges on a graph examples of applications to periodicity location shift detection kolmogorovsmirnov tests and outlier detection are given

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

fractal just infinite nil lie superalgebra of finite width abstract the grigorchuk and gupta sidki groups play fundamental role in modern group theory they are natural examples of self similar finitely generated periodic groups as their natural analogues there are constructions of nil lie p algebras over a field of characteristic 2 40 and arbitrary positive characteristic 52 in characteristic zero similar examples of lie algebras do not exist by a result of martinez and zelmanov 32 the second author constructed analogues of the grigorchuk and gupta sidki groups in the world of lie superalgebras of arbitrary characteristic the virtue of that construction is that the lie superalgebras have clear monomial bases 41 that lie superalgebras have slow polynomial growth and are graded by multidegree in the generators in particular a self similar lie superalgebra q is z 3 graded by multidegree in 3 generators its z 3 components lie inside an elliptic paraboloid in space the components are at most one dimensional thus the z 3 grading of q is fine an analogue of the periodicity is that homogeneous elements of the grading q q 0 q 1 are ad nilpotent in particular q is a nil finely graded lie superalgebra which shows that an extension of the mentioned result of martinez and zelmanov 32 to the lie superalgebras of characteristic zero is not valid but computations with q are rather technical in this paper we construct a similar but simpler and smaller example namely we construct a 2 generated fractal lie superalgebra r over arbitrary field we find a clear monomial basis of r and unlike many examples studied before we find also a clear monomial basis of its associative hull a the latter has a quadratic growth the algebras r and a are z 2 graded by multidegree in the generators positions of their z 2 components are bounded by pairs of logarithmic curves on plane the z 2 components of r are at most one dimensional thus the z 2 grading of r is fine as an analogue of periodicity we establish that homogeneous elements of the grading r r 0 r 1 are ad nilpotent in case of n graded algebras a close analogue to being simple is being just infinite unlike previous examples of lie superalgebras we are able to prove that r is just infinite but not hereditary just infinite our example is close to the smallest possible example because r has a linear growth with a growth function r m 3 m as m moreover r is of finite width 4 char k 2 in case char k 2 we obtain a lie algebra of width 2 that is not thin thus we have got a more handy analogue of the grigorchuk and gupta sidki groups the constructed lie superalgebra r is of linear growth of finite width 4 and just infinite it also shows that an extension of the result of martinez and zelmanov 32 to the lie superalgebras of characteristic zero is not valid

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

free subalgebras of graded algebras let k be a field and let a n 1an be a positively graded k algebra we recall that a is graded nilpotent if for every d 1 the subalgebra of a generated by elements of degree d is nilpotent we give a method of producing grading nilpotent algebras and use this to prove that over any base field k there exists a finitely generated graded nilpotent algebra that contains a free k subalgebra on two generators

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

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

an infinite dimensional affine nil algebra with finite gelfand kirillov dimension the famous 1960 s construction of golod and shafarevich yields infinite dimensional nil but not nilpotent algebras however these algebras have exponential growth here we construct an infinite dimensional nil but not locally nilpotent algebra which has polynomially bounded growth

golod shafarevich groups with property t and kac moody groups we construct golod shafarevich groups with property t and thus provide counterexamples to a conjecture stated in a recent paper of zelmanov ze2 explicit examples of such groups are given by lattices in certain topological kac moody groups over finite fields we provide several applications of this result including examples of residually finite torsion non amenable groups

rejecton of a single outiier in two or three way layouts accurate bounds are presented for the fractiles of the maximum normed residual which is often used to test for a single outlier for two and three way layouts it is shown that the second bonferroni bound of the critical value while not conservative is an excellent approximation to the critical value being much more accurate than the first bonferroni upper bound the third bonferroni upper bound which although conservative is expensive to calculate agrees with the second bound to at least four decimal places for all factor combinations considered in this paper

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

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

existence of multiple solutions for quasilinear systems via fibering method existence and multiplicity results for a general class of radial quasilinear equations are proved using the fibering method introduced and developed by s i pohozaev

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

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

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

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

free subalgebras of division algebras over uncountable fields we study the existence of free subalgebras in division algebras and prove the following general result if a is a noetherian domain which is countably generated over an uncountable algebraically closed field k of characteristic 0 then either the quotient division algebra of a contains a free algebra on two generators or it is left algebraic over every maximal subfield as an application we prove that if k is an uncountable algebraically closed field and a is a finitely generated k algebra that is a domain of gk dimension strictly less than 3 then either a satisfies a polynomial identity or the quotient division algebra of a contains a free k algebra on two generators

non radial solutions for a non homogeneous henon equation abstract in this paper we study a henon like equation see eq 1 where the nonlinearity f t is not homogeneous i e it is not a power by minimization on the nehari manifold we prove that for large values of the parameter there is a breaking of symmetry and non radial solutions appear this holds for sub and super critical growth of the nonlinearity f

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

on a class of nonhomogeneous equations of henon type symmetry breaking and non radial solutions abstract in this work we study the following henon type equation div u p 2 u x a p x f u in b u 0 in b u 0 on b where b x r n x 1 is a ball centered at the origin the parameters verify the inequalities 0 a n p p n 4 0 2 p n p p n p a 1 and the nonlinearity f is nonhomogeneous by minimization on the nehari manifold we prove that for large values of the parameter there is a symmetry breaking and non radial solutions appear

the significance attained by the best fitting regressor variable abstract in the forward selection of independent variables for a linear regression model the next best fitting variable for inclusion is the variable with largest f to enter we consider the evaluation of the p value for this selection

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

polynomial rings over nil rings need not be nil abstract we construct a nil algebra over a countable field the polynomial ring over which is not nil this answers a question of amitsur

existence of positive solutions for the henon equation involving critical sobolev terms abstract let n 3 2 2 n n 2 and r n be a bounded domain with a smooth boundary and 0 our purpose in this paper is to consider the existence of solutions of henon equation u x x u x 2 1 in u 0 in u 0 on where 0

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

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