on groups that have normal forms computable in logspace abstract we consider the class of finitely generated groups which have a normal form computable in logspace we prove that the class of such groups is closed under passing to finite index subgroups direct products wreath products and certain free products and infinite extensions and includes the solvable baumslag solitar groups as well as non residually finite and hence non linear examples we define a group to be logspace embeddable if it embeds in a group with normal forms computable in logspace we prove that finitely generated nilpotent groups are logspace embeddable it follows that all groups of polynomial growth are logspace embeddable

the word and geodesic problems in free solvable groups we study the computational complexity of the word problem wp in free solvable groups s r d where r 2 is the rank and d 2 is the solvability class of the group it is known that the magnus embedding of s r d into matrices provides a polynomial time decision algorithm for wp in a fixed group s r d unfortunately the degree of the polynomial grows together with d so the uniform algorithm is not polynomial in d in this paper we show that wp has time complexity o rn log 2 n in s r 2 and o n 3 rd in s r d for d 3 however it turns out that a seemingly close problem of computing the geodesic length of elements in s r 2 is np complete we prove also that one can compute fox derivatives of elements from s r d in time o n 3 rd in particular one can use efficiently the magnus embedding in computations with free solvable groups our approach is based on such classical tools as the magnus embedding and fox calculus as well as on relatively new geometric ideas in particular we establish a direct link between fox derivatives and geometric flows on cayley graphs

sublinear deviation between geodesics and sample paths we give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic like properties as an application we prove sublinear tracking in teichmuller distance for random walks on mapping class groups and on cayley graphs of a large class of finitely generated groups

growth series of some wreath products the growth series of certain finitely generated groups which are wreath products are investigated these growth series are intimately related to the traveling salesman problem on certain graphs a large class of these growth series is shown to consist of irrational algebraic functions n 0 since only one generating set will be associated with each group below the generating set associated with fr x will be obvious let h be a group with a finite generating set sh these will be fixed for the rest of the paper the cayley graph of the pair h sh is as usual the directed graph whose vertices are the elements of h and there is an edge from a vertex hx xoa vertex h2 if and only if h2 hxh for some h in sh us 1 in particular the edge from hx to h2 has an opposite edge from h2 to hx let c be the graph gotten from the cayley graph of h sh simply by identifying opposite edges in other words c might be called the undirected cayley graph of h sh hex k also be a group with a finite generating set sk these will also be fixed for the rest of the paper it is possible to form what might be called a restricted direct product group p which consists of all functions p from the vertices of c to k such that there are only finitely many vertices v in c with p v t 1 this is a subgroup of the direct product group whose elements consist of all functions from the vertex set of c to k the group p admits h as a group of automorphisms by means of the action of h on c the resulting semidirect product p x h is the restricted wreath product k h

self adaptive operator splitting methods for monotone variational inequalities solving a variational inequality problem vi f is equivalent to finding a solution of a system of nonsmooth equations a hard problem the peaceman rachford and or douglas rachford operator splitting methods are advantageous when they are applied to solve variational inequality problems because they solve the original problem via solving a series of systems of nonlinear smooth equations a series of easy problems although the solution of vi f is invariant under multiplying f by some positive scalar yet the numerical experiment has shown that the number of iterations depends significantly on the positive parameter which is a constant in the original operator splitting methods in general it is difficult to choose a proper parameter for individual problems in this paper we present a modified operator splitting method which adjusts the scalar parameter automatically per iteration based on the message of the iterates exact and inexact forms of the modified method with self adaptive variable parameter are suggested and proved to be convergent under mild assumptions finally preliminary numerical tests show that the self adaptive adjustment rule is proper and necessary in practice

rational and transcendental growth series for the higher heisenberg groups this paper considers growth series of 2 step nilpotent groups with infinite cyclic derived subgroup every such group g has a subgroup of finite index of the form h n m where h n is the discrete heisenberg group of length 2n 1 we call n the heisenberg rank of g we show that every group of this type has some finite generating set such that the corresponding growth series is rational on the other hand we prove that if g has heisenberg rank n 2 then g possesses a finite generating set such that the corresponding growth series is a transcendental power series

numerical methods for the stokes and navier stokes equations driven by threshold slip boundary conditions abstract in this article we discuss the numerical solution of the stokes and navier stokes equations completed by nonlinear slip boundary conditions of friction type in two and three dimensions to solve the stokes system we first reduce the related variational inequality into a saddle point point problem for a well chosen augmented lagrangian to solve this saddle point problem we suggest an alternating direction method of multiplier together with finite element approximations the solution of the navier stokes system combines finite element approximations time discretization by operator splitting and augmented lagrangian method numerical experiment results for two and three dimensional flow confirm the interest of these approaches

solving linear variational inequality problems by a self adaptive projection method we propose a self adaptive projection method for solving linear variational inequality problems and show its global convergence under mild conditions some numerical results are also addressed which indicate that the method is quite robust and efficient

two projection methods for the solution of signorini problems abstract two iterative methods based on projection and boundary element methods are considered for signorini problems the regularized problem with the projection boundary condition is first deduced from the signorini problem by using the equivalence between the signorini boundary condition and the projection fixed point problem our methods formulate the signorini boundary condition into a sequence of robin boundary conditions in the new boundary condition there is a penalty parameter the convergence speed of the first method is greatly influenced by the value of which is difficult to choose for individual problems to improve the performance of this method we present a self adaptive projection method which adjusts the parameter automatically per iteration based on the iterative data the main result of this work is to provide the convergence of the methods under mild assumptions as the iteration process is given by the potential and its derivative on the boundary of the domain the unknowns of the problem are computed explicitly by using the boundary element method both theoretical results and numerical experiments indicate efficiency of the methods proposed

analysis of uncertain structural systems using interval analysis the imprecision or uncertainty present in many engineering analysis design problems can be modeled using probabilistic fuzzy or interval methods this work considers the modeling of uncertain structural systems using interval analysis by representing each uncertain input parameter as an interval number a static structural analysis problem can be expressed in the form of a system of linear interval equations in addition to the direct and gaussian elimination based solution approaches a combinatorial approach based on an exhaustive combination of the extreme values of the interval numbers and an inequality hased method are presented for finding the solution of interval equations the range or interval of the solution vector response parameters is found to increase with increasing size of the problem in all of the methods an interval truncation approach is proposed to limit the growth of intervals of response parameters so that realistic and accurate solutions can be obtained in the presence of large amounts of uncertainty numerical examples are presented to illustrate the computational aspects of the methods and also to indicate the importance of the truncation approach in practical problems the utility of interval methods in predicting the extreme values of the response parameters of structures is discussed

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

an adaptation of nitscheʼs method to the tresca friction problem abstract we propose a simple adaptation to the tresca friction case of the nitsche based finite element method introduced previously for frictionless unilateral contact both cases of unilateral and bilateral contact with friction are taken into account with emphasis on frictional unilateral contact for the numerical analysis we manage to prove theoretically the fully optimal convergence rate of the method in the h 1 norm which is o h 1 2 when the solution lies in h 3 2 0 k 1 2 in two dimensions and three dimensions for lagrange piecewise linear k 1 and quadratic k 2 finite elements no additional assumption on the friction set is needed to obtain this proof

existence of the solution to stationary navier stokes equations with nonlinear slip boundary conditions abstract the stationary navier stokes equations with nonlinear slip boundary conditions are investigated in this paper because the boundary conditions include the subdifferential property on the part boundary the variational formulation of this problem is the variational inequality problem of the second kind with navier stokes operator the main purpose of the paper is to study the existence of the weak solution and the strong solution to this variational inequality problem in terms of the yosida s regularity method

some developments in general variational inequalities general variational inequalities provide us with a unified natural novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences in this paper we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection wiener hopf equations updating the solution auxiliary principle inertial proximal penalty function dynamical system and well posedness we also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection type algorithms our proofs of convergence are very simple as compared with other methods our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems since the general variational inequalities include quasi variational inequalities and quasi implicit complementarity problems as special cases results presented here continue to hold for these problems several open problems have been suggested for further research in these areas

growth and ergodicity of context free languages a language l over a finite alphabet e is called growth sensitive if forbidding any set of subwords f yields a sublanguage l f whose exponential growth rate is smaller than that of l it is shown that every ergodic unambiguous nonlinear context free language is growth sensitive ergodic means for a context free grammar and language that its dependency di graph is strongly connected the same result as above holds for the larger class of essentially ergodic context free languages and if growth is considered with respect to the ambiguity degrees then the assumption of unambiguity may be dropped the methods combine a construction of grammars for 2 block languages with a generating function technique regarding systems of algebraic equations

proper actions of wreath products and generalizations we study stability properties of the haagerup property and of coarse embeddability in a hilbert space under certain semidirect products in particular we prove that they are stable under taking standard wreath products our construction also allows for a characterization of subsets with relative property t in a standard wreath product

fuzzy finite element approach for the analysis of imprecisely defined systems many engineering systems are too complex to be defined in precise mathematical terms they often contain information and features that are vague imprecise qualitative linguistic or incomplete the traditional deterministic and probabilistic techniques are not adequate to analyze such systems this paper aims at developing a fuzzy finite element approach for the analysis of imprecisely defined systems the development of the methodology starts from the basic concepts of fuzzy numbers of fuzzy arithmetic and implements suitably defined fuzzy calculus concepts such as differentiation and integration for the derivation manipulation and solution of the finite element equations simple stress analysis problems involving vaguely defined geometry material properties external loads and boundary conditions are solved to establish and to illustrate the new procedure the approach developed is applicable to systems that are described in linguistic terms as well as those that are described by incomplete information if complete data are known the method handles the information similar to that of a probabilistic approach the present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures

a projection method based on self adaptive rules for stokes equations with nonlinear slip boundary conditions abstract in this work we propose and study a self adaptive projection method for the numerical solution of stokes equations under slip boundary conditions we introduce the projection operator to formulate the slip constraint as a fixed point equation to solve this problem we suggest a projection method by reformulating the nonlinear slip boundary conditions into an iterative form to make the method more efficient we find a self adaptive rule that uses iterative functions to adjust the penalty parameter automatically we show the convergence of the method in function space and give its application in detail finally the numerical results are given to support our theoretical results

c graph automatic groups abstract we generalize the notion of a graph automatic group introduced by kharlampovich khoussainov and miasnikov by replacing the regular languages in their definition with more powerful language classes for a fixed language class c we call the resulting groups c graph automatic we prove that the class of c graph automatic groups is closed under change of generating set direct and free product for certain classes c we show that for quasi realtime counter graph automatic groups where normal forms have length that is linear in the geodesic length there is an algorithm to compute normal forms and therefore solve the word problem in polynomial time the class of quasi realtime counter graph automatic groups includes all baumslag solitar groups and the free group of countably infinite rank context sensitive graph automatic groups are shown to be a very large class which encompasses for example groups with unsolvable conjugacy problem the grigorchuk group and thompson s groups f t and v

projection and self adaptive projection methods for the signorini problem with the bem abstract we propose two new projection methods for solving the signorini problem and study the relationship between them since the signorini boundary condition is equivalent to a projection fixed point problem the methods formulate the signorini boundary condition into a sequence of robin boundary conditions and only require solving a variational problem with simple boundary conditions at each iterative step the convergence speed of the first method depends on the parameter heavily and it is difficult to choose a proper for individual problems to improve the efficiency of the method we propose an adaptive projection method which adjusts the parameter automatically per iteration we establish the convergence analysis of the methods because the signorini boundary condition is given in an iterative form by the potential and its derivative on the boundary of the domain we can easily provide the boundary element approximation of the problem finally we present some numerical simulations which illustrate the performance of the methods