the weil petersson hessian of length on teichmuller space we present a brief but nearly self contained proof of a formula for the weil petersson hessian of the geodesic length of a closed curve either simple or not simple on a hyperbolic surface the formula is the sum of the integrals of two naturally defined positive functions over the geodesic proving convexity of this functional over teichmuller space due to wolpert 1987 we then estimate this hessian from below in terms of local quantities and distance along the geodesic the formula extends to proper arcs on punctured hyperbolic surfaces and the estimate to laminations wolpert s result that the thurston metric is a multiple of the weil petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves we give further applications to upper bounds of the hessian especially near pinching loci recover through a geometric argument wolpert s result on the convexity of length to the half power and give a lower bound for growth of length in terms of twist

a partial analogue of borel s fixed point theorem for finite p groups abstract borel s fixed point theorem states that a solvable connected algebraic group g on a non empty complete variety v must have a fixed point thus if v consists of subgroups of g and g acts on v by conjugation then some subgroup in v is normal in g although g is infinite or trivial in borel s theorem the method of proof yields applications to finite p groups they show that in many cases a family of subgroups of a finite p group must contain at least one normal subgroup here we obtain some applications of this type

the classification of endo trivial modules this paper settles the classification of all endo trivial modules for a finite p group by treating the remaining case of a p group having maximal elementary abelian subgroups of rank 2 it is shown that the modules constructed by alperin using relative syzygies provide the complete list of endo trivial modules in this case

the class breadth conjecture revisited abstract the class breadth conjecture for groups with prime power order was formulated by leedham green neumann and wiegold in 1969 we construct a new counter example to the conjecture it has order 2 19 and is a quotient of a 4 dimensional 2 uniserial space group we translate the conjecture to p uniserial space groups prove that these have finite cobreadth and provide an explicit upper bound we develop an algorithm to decide the conjecture for p uniserial space groups and use this to show that all 3 uniserial space groups of dimension at most 54 satisfy the conjecture we show that over every finite field there are lie algebras which fail the corresponding conjecture

length of a curve is quasi convex along a teichmuller geodesic we show that for every simple closed curve alpha the extremal length and the hyperbolic length of alpha are quasi convex functions along any teichmuller geodesic as a corollary we conclude that in teichmuller space equipped with the teichmuller metric balls are quasi convex

coincidence theorems and variational inequalities for fuzzy mappings abstract in this paper some coincidence theorems and existence theorems of solutions of variational inequalities for fuzzy mappings are established

coincidence degree and coincidence theorems for fuzzy mappings abstract in this paper the concept of coincidence degree and some coincidence theorems for fuzzy mappings have been introduced and established the results presented in this paper generalize and unify some results of 2 3 4 5 8 and give a fuzzy generalization of the famous kakutani ky fan theorem

fixed degree for fuzzy mappings and a generalization of ky fan s theorem abstract in this paper the concept of some properties of fixed degree for fuzzy mappings are introduced and studied the results presented generalize and unify some recent results

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

metabelian thin lie algebras abstract a graded lie algebra is thin if it is generated by two elements of degree 1 and each of its homogeneous ideals is located between two consecutive terms of the lower central series in this paper we give a complete classification of the metabelian thin lie algebras and their graded automorphism groups

restricted one dimensional central extensions of the restricted filiform lie algebras m0λ p abstract consider the filiform lie algebra m 0 with nonzero lie brackets e 1 e i e i 1 for 1 i p where the characteristic of the field f is p 0 we show that there is a family m 0 p of restricted lie algebra structures parameterized by elements f p we explicitly describe both the ordinary and restricted 1 cohomology spaces and show that for p 3 these spaces are equal we also describe the ordinary and restricted 2 cohomology spaces and interpret our results in the context of one dimensional central extensions

fusion systems containing pearls abstract an f essential subgroup is called a pearl if it is either elementary abelian of order p 2 or non abelian of order p 3 in this paper we start the investigation of fusion systems containing pearls we determine a bound for the order of p groups containing pearls and we classify the saturated fusion systems on p groups containing pearls and having sectional rank at most 4

a generalization of generalized quasi variational inequalities abstract the aim of this paper is to give an application of a fixed point theorem of a compact acyclic multifunction to a generalized quasi variational inequality problem in a non compact setting we present some general acyclic valued versions of several known results due to kim shih and tan

thin lie algebras with diamonds of finite and infinite type we investigate a class of infinite dimensional modular lie algebras graded over the positive integers in which every homogeneous component has dimension one or two we identify these lie algebras with loop algebras of certain simple lie algebras of hamiltonian type these lie algebras are not finitely presented but certain central extensions of them are and we give an explicit construction for the latter

a family of simple lie algebras in characteristic two abstract in this paper a new two parameter family of simple lie algebras defined over fields of characteristic two is described they are used to construct loop algebras which are central to the classification of graded lie algebras with maximal class and characteristic two

graded lie algebras of maximal class iii we describe the isomorphism classes of infinite dimensional n graded lie algebras of maximal class generated by their first homogeneous component over fields of characteristic two this complements the analogous work by caranti and newman in the odd characteristic case

narrow lie algebras a coclass theory and a characterization of the witt algebra abstract in this paper we examine some narrowness conditions for lie algebras over a fieldfof characteristic zero in particular we show that the natural analogs of the main coclass conjectures forp groups hold in the context of n graded lie algebraslwhich are generated by their first homogeneous componentl1 while lie algebras of finite coclass need not be soluble we show that the positive part of the witt algebra der f x is the only non soluble n graded lie algebralof coclass 1 which is generated byl1andl2

thin loop algebras of albert zassenhaus algebras thin lie algebras are lie algebras over a field graded over the positive integers and satisfying a certain narrowness condition in particular all homogeneous components have dimension one or two and are called diamonds in the latter case the first diamond is the component of degree one and the second diamond can only occur in degrees 3 5 q or 2q 1 where q is a power of the characteristic of the underlying field here we consider several classes of thin lie algebras with second diamond in degree q in particular we identify the lie algebras in one of these classes with suitable loop algebras of certain albert zassenhaus lie algebras we also apply a deformation technique to recover other thin lie algebras previously produced as loop algebras of certain graded hamiltonian lie algebras

ideally constrained lie algebras abstract in this paper we deal with graded lie algebras l such that there exists a positive integer r such that for every positive integer i and for every homogeneous ideal i l i the inclusion i l i r 1 holds the solvable case and the r 1 case receive a special attention

fixed points for kakutani factorizable multifunctions abstract a multifunction is called a kakutani multifunction if there exist two nonempty convex sets x and y each in a hausdorff topological vector space such that x y is upper semi continuous with nonempty compact convex values we prove the following extension of the kakutani fixed point theorem let x x be a multifunction from a simplex x into itself if can be factorized by an arbitrary finite number of kakutani multifunctions then has a fixed point the proof relies on a simplicial approximation technique and the brouwer fixed point theorem extensions to infinite dimensional spaces and applications to game theory are given

endotrivial modules for p solvable groups we determine the torsion subgroup of the group of endotrivial modules for a finite solvable group in characteristic p we also prove that our result would hold for p solvable groups provided a conjecture can be proved about the case of p nilpotent groups

the nilpotency class of p groups in which subgroups have few conjugates abstract we consider the problem of how the nilpotency class of a finite p group can be bounded in terms of the maximum length of the conjugacy classes of cyclic subgroups we sharpen some previously known bounds and also prove that a p group in which every cyclic subgroup has at most p 2 conjugates has class at most 4

connected components of the category of elementary abelian p subgroups we determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p group g for an odd prime p namely it is p if g has rank at least 3 and it is p 1 if g has rank 2 more precisely if g has rank 2 there are exactly 1 2 p 1 or possibly 3 classes for some 3 groups of maximal nilpotency class

fixed point theorems for fuzzy mappings abstract this paper presents some new fixed point theorems for fuzzy mappings the results given in this paper improve and extend some recent results

p groups with maximal elementary abelian subgroups of rank 2 abstract let p be an odd prime number and g a finite p group we prove that if the rank of g is greater than p then g has no maximal elementary abelian subgroup of rank 2 it follows that if g has rank greater than p then the poset e g of elementary abelian subgroups of g of rank at least 2 is connected and the torsion free rank of the group of endotrivial kg modules is one for any field k of characteristic p we also verify the class breadth conjecture for the p groups g whose poset e g has more than one component

graded lie algebras of maximal class ii abstract we describe the isomorphism classes of infinite dimensional n graded lie algebras of maximal class over fields of odd characteristic generated by their first homogeneous component

graded lie algebras of maximal class we study graded lie algebras of maximal class over a field mathbf f of positive characteristic p a shalev has constructed infinitely many pairwise non isomorphic insoluble algebras of this kind thus showing that these algebras are more complicated than might be suggested by considering only associated lie algebras of p groups of maximal class here we construct mathbf f aleph 0 pairwise non isomorphic such algebras and max mathbf f aleph 0 soluble ones both numbers are shown to be best possible we also exhibit classes of examples with a non periodic structure as in the case of groups two step centralizers play an important role

limits of abelian subgroups of finitep groups abelian subgroups play a key role in the theory and applications of finite p groups our purpose is to establish some very general results motivated by special results that have been of use in particular it is w x known kj that if a finite p group for odd p has an elementary abelian subgroup of order p n f 5 then it has a normal elementary abelian subgroup of the same order our first main result is a general one of this type

sharpened forms of coincidence theorems and variational inequalities for fuzzy mappings abstract the aim of this paper is to generalize and to sharpen recent results of chang 1994 on coincidence point and variational inequalities for fuzzy mappings our generalizations are summarized as follows to begin with we remove some essential conditions which chang s results cannot do without secondly we provide very simple proofs for those results of chang finally we extend chang s theorem on coincidence points between to spaces to the case of n 2 spaces in addition we use non compact settings all the above refinements can be simultaneously achieved with the aid of some known results due to kum 1994 and simons 1986

measured foliations and the minimal norm property for quadratic differentials the objective is to prove two important inequalities of riemann surface theory and to present them in such a way that one can see their close relationship the first inequality is the minimal norm property of quadratic differentials first observed by marden and strebel 8 actually a somewhat different inequality is given the minimum norm properties have many applications one of which is a proof of the result of hubbard and masur 7 concerning the existence of a unique holomorphic quadratic differential whose horizontal foliation and vertical measure realize a given measure class of measured foliations on a given riemann surface this result is given in section 7 the method of proof is quite different from the one in 7 the second important inequality is the main inequality of reich and strebel it has central importance in teichmiiller theory and can be used to derive the infinitesimal form of teichmiiller s metric 10 moreover this inequality can be used to show that teichmiiller s metric is the integral of its infinitesimal form 6 a proof of this result by other methods was given by o byrne in 9 both of these inequalities turn out to be consequences of certain properties of the trajectory structure of a holomorphic quadratic differential for the purposes of the main inequality we are able to bypass some of the theory of trajectories by means of an averaging device used by teichmiiller this averaging device also appears in the proofs of teichmiiller s theorem given by bers 4 and abikoff 1 but for the minimum norm property in the form given by marden and strebel 8 the detailed theory of trajectories seems to be necessary