algorithmically insoluble problems about finitely presented solvable groups lie and associative algebras 1 1 in 1955 novikov 111 constructed a finitely presented group with an insoluble word problem cf miller lo and also boone 8 shortly afterwards first adyan l 4 and then rabin 13 143 used this group of novikov to prove that for every one of a host of group theoretic properties there is no algorithm whereby one can decide whether any finitely presented group has such a property in particular there is no algorithm whereby one can decide whether any finitely presented group is of order 1 thus the isomorphism problem for finitely presented groups is algorithmically insoluble these negative results were followed in 1959 by a number of similar ones concerned with elements and subgroups of finitely presented groups baumslag boone and neumann s thus for example there is no algorithm whereby one can decide whether or not any element in a suitably chosen finitely presented group is a proper power in 1981 harlampovich 9 constructed a finitely presented solvable group of derived length 3 with an insoluble word problem this then raised the possibility that many of the results cited above for finitely presented groups can be carried over to finitely presented solvable groups it is not possible to carry over these results en masse to finitely presented solvable groups since it is easy to decide whether any finitely presented solvable group is of order 1 indeed there is an algorithm whereby one can determine whether any finitely presented solvable group is polycyclic baumslag cannonito and miller 6 again no such algorithm exists for finitely presented groups as a whole adyan 2 the methods employed previously cannot be applied to solvable groups because the constructions

zeros of orthogonal polynomials in certain discrete sobolev spaces abstract in this work discrete sobolev pseudo inner products of type 1 p q p c q c a b p x q x d x where d is a quasi definite borel measure and 0 and 2 p q p c q c p c q c a a p x q x d x where d is a symmetric positive borel measure c 0 and 0 are considered general properties of orthogonal polynomials associated with the above discrete sobolev inner products and their zeros are studied

testing polycyclicity of finitely generated rational matrix groups we describe algorithms for testing polycyclicity and nilpotency for finitely generated subgroups of gl d q and thus we show that these properties are decidable variations of our algorithm can be used for testing virtual polycyclicity and virtual nilpotency for finitely generated subgroups of gl d q

algorithms for the tits alternative and related problems abstract we present an algorithm to decide whether a finitely generated linear group over an infinite field is solvable by finite thereby obtaining a computationally effective version of the tits alternative we also give algorithms to decide whether the group is nilpotent by finite abelian by finite or central by finite implementations of the algorithms are publicly available in magma

convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems abstract the purpose of this work is to introduce a hybrid projection method for finding a common element of the set of a generalized equilibrium problem the set of solutions to a variational inequality and the set of fixed points of a strict pseudo contraction in a real hilbert space

an iterative method of solution for equilibrium and optimization problems abstract in this paper we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping the set of solutions of the variational inequality for an inverse strongly monotone mapping and the set of solutions of an equilibrium problem in a hilbert space we show that the iterative sequence converges strongly to a common element of the three sets the results of this paper extended and improved the results of h iiduka and w takahashi strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings nonlinear anal 61 2005 341 350 and s takahashi and w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2007 506 515 therefore by using the above result an iterative algorithm for the solution of a optimization problem was obtained

computable algebra and group embeddings this work grew out of our investigations of the infinitely generated subgroups of finitely presented groups a useful technique for embedding countable groups with sufficiently nice local properties into finitely presented groups was introduced in l this technique was then applied in 1 21 to prove that every countable locally polycyclic group and every countable metabelian group can be embedded in a finitely presented group it became apparent from 2 that effective methods in commutative algebra could be extended to group rings of polycyclic groups and then used to embed certain groups in finitely presented groups in this paper these extensions are carried out with the consequent application that every countable locally abelian by nilpotent group can be embedded in a finitely presented group much of the effective commutative algebra we require seems to be known see for instance 6 9 14 171 unfortunately a systematic account is not available in particular we need a proof of the hilbert basis theorem which

viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications abstract in this paper we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems the set of common fixed points of a family of infinite non expansive mappings strong convergence theorems are established in a real hilbert space under suitable conditions as some applications we consider convex feasibility problems and equilibrium problems the results presented improve and extend the corresponding results of many others

the algorithmic theory of polycyclic by finite groups a group is said to be polycyclic by finite or a pf group for short if it has a polycyclic normal subgroup of finite index equivalently pf groups are exactly the groups which have a series of finite length whose infinite factors are cyclic by a well known theorem of p hall every pf group is finitely presented and in fact pf groups form the largest known sectionclosed class of finitely presented groups it is this fact that makes pf groups natural objects of study from the algorithmic standpoint the general aim of the algorithmic theory of pf groups can be described as the collection of information about a pf group which can in principle

some new algorithms for solving mixed equilibrium problems in this paper we suggest and analyze two projection methods one implicit and one explicit for finding a particular solution of a mixed equilibrium problem in a real hilbert space furthermore we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem

convergence theorems of common elements for equilibrium problems and fixed point problems in banach spaces the purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of two quasi f nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of banach spaces our results improve and extend the corresponding results announced by many others

symmetrical orthogonal polynomials for sobolev type inner products abstract in this paper families of symmetric orthogonal polynomials qn with respect to the sobolev type inner product f g ifgd rj 0mj j 0 g j 0 where i is a symmetric interval and is a symmetric positive borel measure with infinite support on i and whose moments are all finite are considered if q2n x un x2 and q2n 1 x xvn x2 we deduce that un and vn are sobolev type orthogonal polynomials and in several particular cases standard orthogonal polynomials we study the zeros of qn showing that in some cases qn has two complex conjugate zeros moreover a partial result about separation of the zeros is given we also discuss the symmetrization problem for this kind of inner products finally some sobolev type inner products with two symmetric mass points are considered

a new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings abstract in this paper we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for inverse strongly monotone mappings we show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions this main theorem extends a recent result of yao and yao y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings applied mathematics and computation 186 2 2007 1551 1558

hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems in this paper we introduce a new iterative algorithm by hybrid method for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of a general variational inequality problem for finite inverse strongly monotone mappings and the set of common fixed points of infinite family of strictly pseudocontractive mappings in a real hilbert space then we prove strong convergence of the scheme to a common element of the three above described sets our result improves and extends the corresponding results announced by many others

deciding isomorphy using dehn fillings the splitting case we solve dehn s isomorphism problem for virtually torsion free relatively hyperbolic groups with nilpotent parabolic subgroups we do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups namely the isomorphism problem separation of torsion in their outer automorphism groups by congruences and the mixed whitehead problem an automorphism group orbit problem the first step of the reduction is to compute canonical jsj decompositions dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if the graphs of groups have isomorphic vertex groups and if so whether a global isomorphism can be assembled for the class of finitely generated nilpotent groups we give solutions to these algorithmic problems by using the arithmetic nature of these groups and of their automorphism groups

asymptotics for varying discrete sobolev orthogonal polynomials abstract we consider a varying discrete sobolev inner product such as f g s f x g x d m n f j c g j c where is a finite positive borel measure supported on an infinite subset of the real line c is adequately located on the real axis j 0 and mn n 0 is a sequence of nonnegative real numbers satisfying a very general condition our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this sobolev inner product in this way we focus our attention on mehler heine type formulae as they describe in detail the asymptotic behavior of these polynomials around c just the point where we have located the perturbation of the standard inner product moreover we pay attention to the asymptotic behavior of the scaled zeros of these varying sobolev polynomials and some numerical experiments are shown finally we provide other asymptotic results which strengthen the idea that mehler heine asymptotics describe in a precise way the differences between sobolev orthogonal polynomials and standard ones

some properties of zeros of sobolev type orthogonal polynomials abstract for polynomials orthogonal with respect to a discrete sobolev product we prove that for each n q n has at least n m zeros on the convex hull of the support of the measure where m denotes the number of terms in the discrete part interlacing properties of zeros are also described

connection formulas for general discrete sobolev polynomials in this paper the discrete sobolev inner product p q p x q x d i 0 r m i p i c q i c is considered where is a finite positive borel measure supported on an infinite subset of the real line c r and mi 0 i 0 1 r connection formulas for the orthonormal polynomials associated with are obtained as a consequence for a wide class of measures we give the mehler heine asymptotics in the case of the point c is a hard edge of the support of in particular the case of a symmetric measure is analyzed finally some examples are presented

strong convergence theorems for an infinite family of quasi nonexpansive mappings and generalized equilibrium problems and variational inequality problems in this paper we introduce a new iterative scheme for finding a common element of the set of common fixed points of an infinite family of uniformly continuous quasi nonexpansive mappings and the set of solutions to a generalized equilibrium problem and the set of solutions to a variational inequality problem in a real hilbert space we then prove strong convergence of the scheme to a common element of the three mentioned sets our results extend important recent results

the algorithmic theory of finitely generated metabelian groups algorithms are constructed which when an explicit presentation of a finitely generated metabelian group g in the variety x 2 is given produce finitary presentations for the derived subgroup g the centre z g the fitting subgroup fit g and the frattini subgroup 0 g additional algorithms of independent interest are developed for commutative algebra which construct the associated set of primes ass m of a finitely generated module m over a finitely generated commutative ring r and the intersection pr m of the maximal submodules of m

iterative methods for strict pseudo contractions in hilbert spaces abstract let t i i 1 n be n strict pseudo contractions defined on a closed convex subset c of a real hilbert space h consider the problem of finding a common fixed point of these mappings and consider the parallel and cyclic algorithms for solving this problem we will prove the weak convergence of these algorithms moreover by applying additional projections we further prove that these algorithms can be modified to have strong convergence

the permutizer condition in infinite soluble groups abstract a group satisfies the permutizer condition p if each proper subgroup permutes with some cyclic subgroup not contained in it here we characterize the classes of soluble minimax groups and finitely generated soluble groups with p

hybrid viscosity like approximation methods for nonexpansive mappings in hilbert spaces consider on a real hilbert space h a nonexpansive mapping t with a fixed point a contraction f with coefficient 0 0 respectively let 0 1 with m n m n and prove the strong convergence of the iterative algorithm to a fixed point x fix t c which is the unique solution of the variational inequality for short v i a i m b cf c 0 x c on the other hand assume c is the intersection of the fixed point sets of a finite number of nonexpansive mappings on h we devise another iterative algorithm which generates a sequence x n from an arbitrary initial point x 0 h the sequence x n is proven to converge strongly to an element of c which is the unique solution x of the v i a i m b cf c applications to constrained generalized pseudoinverses are included

on the orbit stabilizer problem for integral matrix actions of polycyclic groups we present an algorithm to solve the orbit stabilizer problem for a polycyclic group g acting as a subgroup of gl d z on the elements of qd we report on an implementation of our method and use this to observe that the algorithm is practical

on the fitting subgroup of a polycyclic by finite group and its applications abstract we present an algorithm for determining the fitting subgroup of a polycyclic by finite group as applications we describe methods for calculating the centre and the fc centre and for exhibiting the nilpotent by abelian by finite structure of a polycyclic by finite group

the approximation of fixed points of compositions of nonexpansive mappings in hilbert space determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences an algorithm for finding common fixed points of nonexpansive mappings in hilbert space essentially due to halpern is analyzed the main theorem extends wittmann s recent work and partially generalizes a result by lions algorithms of this kind have been applied to the convex feasibility problem