supports of derivations free factorizations and ranks of fixed subgroups in free groups for f a free group of finite rank it is shown that the fixed subgroup of any set b of endomorphisms of f has rank rank f generalizing a recent result of dicks and ventura the proof involves the combinatorics of derivations of groups some related questions are examined

solitary and periodic wave solutions of calogero bogoyavlenskii schiff equation via exp function methods abstract in this paper some proficient techniques called the exp function method modified exp function method and improved modified exp function method are implemented to compute generalized solitary and periodic wave solutions of the nonlinear calogero bogoyavlenskii schiff equation it has been attempted to show the compatibility and reliability of these mathematical techniques the first partial differential equation is converted into an ordinary differential equation by a suitable transformation then a trial solution is assumed results indicate complete consistency and effectiveness of the suggested schemes

fixed subgroups of endomorphisms of free products abstract let g i 1 n g i and let be a symmetric endomorphism of g if is a monomorphism or if g is a finitely generated residually finite group then the fixed subgroup fix g g g g of has kurosh rank at most n

the first integral method to some complex nonlinear partial differential equations in this paper the first integral method is used to construct exact solutions of the hamiltonian amplitude equation and coupled higgs field equation the first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations this method can be applied to nonintegrable equations as well as to integrable ones

examples of retracts in free groups that are not the fixed subgroup of any automorphism let n be an arbitrary cardinal and let fn be a free group of rank n the flxed subgroup of an endomorphismof fn is the subgroup of elements in fn flxed by in this paper the relationship between the family of flxed subgroups of endomorphisms of fn and the family of flxed subgroups of automorphisms of fn will be studied we prove that these two families of subgroups do not coincide for n 3 by showing an inflnite sequence of explicit examples of retracts of fn and so flxed subgroups of endomorphisms of fn which are not flxed subgroups of any automorphism of fn

the group fixed by a family of endomorphisms of a surface group for a closed surface s with s 0 we show that the fixed subgroup of a family b of endomorphisms of 1 s has rankfixb rank 1 s in particular if b contains a non epimorphic endomorphism then rankfixb 12rank 1 s we also show that geometric subgroups of 1 s are inert and hence the fixed subgroup of a family of epimorphisms of 1 s is also inert

on equalizers of sections abstract let g and m be groups and a b g g m group theoretic sections of the natural projection g m g we use the almost stability theorem pro trees and new folding sequence techniques to show that if g is finitely generated then the equalizer of a and b is a free factor of g which answers a question of g m bergman

identifiability of bivariate survival curves from censored data abstract we show that the survival curve is identifiable in bivariate censored data problems under weaker independence assumptions than have commonly been made the common assumption has been mutual independence of t 1 t 2 and z 1 z 2 where t 1 t 2 is the true survival vector z 1 z 2 is a nuisance censoring vector and bivariate right censored data is observed we show that the distribution of t 1 t 2 is identifiable under weaker conditional independence assumptions for distributions with full support bivariate survival analysis is a more powerful analysis tool than univariate analysis if multiple possibly related times are of interest the mutual independence model has become popular as a nonparametric way of analyzing such data analysis of the bivariate problem and analogy with univariate models are used to show that the conditional independence model is more widely applicable as a general nonparametric model for bivariate survival data

embedding rings in completed graded rings 4 commutative algebras abstract if k is a commutative ring of finite characteristic and a a commutative associative k algebra with augmentation such that the intersection of the powers of the augmentation ideal is zero it is shown that a can be embedded in a completed graded commutative associative k algebra with degree 0 component just k the corresponding result is false in characteristic 0 though we do not know whether it can fail when k is a field the above finite characteristic result is also true with commutative associative algebra replaced by any variety of associative or nonassociative algebras defined by multilinear identities

embedding rings in completed graded rings 1 triangular embeddings abstract let k be a commutative ring with 1 r an associative k algebra without 1 and n a positive integer it is shown that the following conditions are equivalent a r can be embedded as a k algebra in the ring of strictly upper triangular n n matrices over an associative k algebra a b r has such an embedding with a further assumed commutative c r can be embedded as a k algebra in a graded associative k algebra c whose only nonzero components are c 1 c nn 1 the proof that the other conditions imply b is essentially a demonstration that a certain kind of k linear category always has a faithful functor into a commutative k algebra subsequent papers in this series determine sufficient conditions for a c above to hold in terms of nilpotence of r under various hypotheses

on automorphism fixed subgroups of a free group abstract let f be a finitely generated free group and let n denote its rank a subgroup h of f is said to be automorphism fixed or auto fixed for short if there exists a set s of automorphisms of f such that h is precisely the set of elements fixed by every element of s similarly h is 1 auto fixed if there exists a single automorphism of f whose set of fixed elements is precisely h we show that each auto fixed subgroup of f is a free factor of a 1 auto fixed subgroup of f we show also that if and only if n 3 then there exist free factors of 1 auto fixed subgroups of f which are not auto fixed subgroups of f a 1 auto fixed subgroup h of f has rank at most n by the bestvina handel theorem and if h has rank exactly n then h is said to be a maximum rank 1 auto fixed subgroup of f and similarly for auto fixed subgroups hence a maximum rank auto fixed subgroup of f is a maximum rank 1 auto fixed subgroup of f we further prove that if h is a maximum rank 1 auto fixed subgroup of f then the group of automorphisms of f which fix every element of h is free abelain of rank at most n 1 all of our results apply also to endomorphisms

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

aspects of free groups if n tuples g h in a rank 2 free group satisfy the same existential formulas then there is an automorphism taking g to h fix a prime p then fk is up to isomorphism the only k generated group having all finite k generated p groups as homomorphic images the theory of non abelian free groups has no prime model

products of test elements of free factors of a free group abstract let g be a group an element g g is called a test element of g if for every endomorphism g g g g implies that is an automorphism let f x be a free group on a finite non empty set x and let x x 1 x 2 x r be a finite partition of x into r 2 non empty subsets for i 1 2 r let u i x i f x and let w z 1 z r be a word in the variables z 1 z r we give several sufficient conditions on u i 1 i r and w for w u 1 u r to be a test element of f x as an application of these results we give examples of test elements of a free group of rank greater than two that are not test elements in any pro p completion of the group

primitive almost primitive test and δ primitive elements of free algebras with the nielsen schreier property abstract we study generalized primitive elements of free algebras of finite ranks with the nielsen schreier property and their automorphic orbits a primitive element of a free algebra is an element of some free generating set of this algebra almost primitive elements are not primitive elements which are primitive in any proper subalgebra primitive elements are elements whose partial derivatives generate the same one sided ideal of the universal multiplicative envelope algebra of a free algebra as the set of free generators generate we prove that an endomorphism preserving an automorphic orbit of a nonzero element of a free algebra of rank two is an automorphism an algorithm to determine test elements of free algebras of rank two is described a series of almost primitive elements is constructed and new examples of test elements are given we prove that if the rank n of the free lie algebra l is even n 2m then any primitive element of l is an automorphic image of the element w x1 x2 x2m 1 x2m there are no primitive elements of l if n is odd and the group of automorphisms of the algebra l acts transitively on the set of all primitive elements

nonparametric estimation of a bivariate survival function in the presence of censoring a new family of estimators of a bivariate survival function based on censored vectors is obtained from a decomposition of the bivariate survival function these estimators are uniformly consistent under bivariate censoring and are self consistent under univariate censoring

the kadomtsev petviashvili equation as a source of integrable model equations a new integrable and nonlinear partial differential equation pde in 2 1 dimensions is obtained by an asymptotically exact reduction method based on fourier expansion and spatiotemporal rescaling from the kadomtsev petviashvili equation the integrability property is explicitly demonstrated by exhibiting the corresponding lax pair that is obtained by applying the reduction technique to the lax pair of the kadomtsev petviashvili equation this model equation is likely to be of applicative relevance because it may be considered a consistent approximation of a large class of nonlinear evolution pdes

estimation of survival curves from dependent censorship models via a generalized self consistent property with nonparametric bayesian estimation application on presente une famille d estimateurs obtenue par generalisation de la propriete autoconsistante efron 1967 de l estimateur de kaplan meier 1958

closed form solutions for coupled nonlinear maccari system abstract the exploration of closed form solutions for nonlinear partial differential equations nlpdes is being an attractive subject in the different branches of mathematical and physical sciences in this paper we have applied extended exp function method to calculate closed form solutions for nlpdes such as the nonlinear maccari system which is very significant in engineering and mathematical physics plentiful closed form solutions with arbitrary parameters are successfully obtained by this method it is shown that the obtained solutions are more general and fresh and can be helpful to analyze the nlpdes in mathematical physics and engineering problems