thompson s group f is not almost convex abstract we show that thompson s group f does not satisfy cannon s almost convexity condition ac n for any positive integer n with respect to the standard generating set with two elements to accomplish this we construct a family of pairs of elements at distance n from the identity and distance 2 from each other which are not connected by a path lying inside the n ball of length less than k for increasingly large k our techniques rely upon fordham s method for calculating the length of a word in f and upon an analysis of the generators geometric actions on the tree pair diagrams representing elements of f

uniform asymptotic regularity for mann iterates abstract in numer funct anal optim 22 2001 641 656 we obtained an effective quantitative analysis of a theorem due to borwein reich and shafrir on the asymptotic behavior of general krasnoselski mann iterations for nonexpansive self mappings of convex sets c in arbitrary normed spaces we used this result to obtain a new strong uniform version of ishikawa s theorem for bounded c in this paper we give a qualitative improvement of our result in the unbounded case and prove the uniformity result for the bounded case under the weaker assumption that c contains a point x whose krasnoselski mann iteration x k is bounded we also consider more general iterations for which asymptotic regularity is known only for uniformly convex spaces groetsch we give uniform effective bounds for an extension of groetsch s theorem which generalize previous results by kirk martinez yanez and the author

on the σ2 invariants of the generalised r thompson groups of type f abstract the 2 invariants of the generalised thompson group f n x 0 x 1 x n x i x j x i n 1 for i j 0 are calculated for n 3 the case n 2 was solved in bieri et al 2010 5

the isomorphism problem for higman thompson groups abstract we prove that the higman thompson groups g n r and g m s are isomorphic if and only if m n and gcd n 1 r gcd n 1 s

spurious solutions in mixed finite element method for maxwell s equations dispersion analysis and new basis functions the finite element method is a well known computational technique used to obtain numerical solutions to boundary value problems including maxwell s equations this paper first presents a brief description of the mathematical structure based on the de rham diagram to discretize maxwell s equations then it uses a numerical dispersion analysis of the mixed finite element method with both electric and magnetic fields as unknowns to evaluate the presence of spurious solutions for different basis functions these unwanted spurious solutions appear when the same order of element is used for electric and magnetic fields while the system is free of spurious modes when different orders of elements are employed for electric and magnetic fields in this work finite elements in both frequency and time domain are studied and the effects of these spurious solutions in both domains are analyzed in one and three dimensional cases

operad groups and their finiteness properties abstract we propose a new unifying framework for thompson like groups using a well known device called operads and category theory as language we discuss examples of operad groups which have appeared in the literature before as a first application we prove a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type f this unifies and extends existing proofs that certain thompson like groups are of type f

groups of piecewise linear homeomorphisms in this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line we use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them we also identify large simple subgroups in all of them providing examples of finitely presented infinite simple groups

cat k spaces weak convergence and fixed points abstract in this paper we show that some of the recent results on fixed point for cat 0 spaces still hold true for cat 1 spaces and so for any cat k space under natural boundedness conditions we also introduce a new notion of convergence in geodesic spaces which is related to the convergence and applied to study some aspects on the geometry of cat 0 spaces at this point two recently posed questions in w a kirk b panyanak a concept of convergence in geodesic spaces nonlinear anal 68 12 2008 3689 3696 are answered in the negative the work finishes with the study of the lifsic characteristic and property p of lim xu to derive fixed point results for uniformly lipschitzian mappings in cat k spaces a conjecture raised in s dhompongsa w a kirk b sims fixed points of uniformly lipschitzian mappings nonlinear anal 65 2006 762 772 on the lifsic characteristic function of cat k spaces is solved in the positive

diffeomorphism groups of critical regularity let m be a circle or a compact interval and let alpha k tau ge 1 be a real number such that k lfloor alpha rfloor we write mathrm diff alpha m for the group of orientation preserving c k diffeomorphisms of m whose kth derivatives are holder continuous with exponent tau we prove that there exists a continuum of isomorphism types of finitely generated subgroups g le mathrm diff alpha m with the property that g admits no injective homomorphisms into bigcup beta alpha mathrm diff beta m we also show the dual result there exists a continuum of isomorphism types of finitely generated subgroups g of bigcap beta alpha mathrm diff beta m with the property that g admits no injective homomorphisms into mathrm diff alpha m the groups g are constructed so that their commutator groups are simple we give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms for example we show that if alpha ge 1 is a real number not equal to 2 then there is no nontrivial homomorphism mathrm diff alpha s 1 rightarrow bigcup beta alpha mathrm diff beta s 1 finally we obtain an independent result that the class of finitely generated subgroups of mathrm diff 1 m is not closed under taking finite free products

free products in r thompson s group v we investigate some product structures in r thompson s group v primarily by studying the topological dynamics associated with v s action on the cantor set c we draw attention to the class d v c of groups which have embeddings as demonstrative subgroups of v which class can be used to assist in forming various products note that d v c contains all finite groups the free group on two generators and q z and is closed under passing to subgroups and under taking direct products of any member by any finite member if g v and h d v c then g oh embeds into v finally if g h d v c then g h embeds in v using a dynamical approach we also show the perhaps surprising result that z z does not embed in v even though v has many embedded copies of z and has many embedded copies of free products of various pairs of its subgroups

some logical metatheorems with applications in functional analysis in previous papers we have developed proof theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis here uniform means independence from parameters in compact spaces a recent case study in fixed point theory systematically yielded uniformity even w r t parameters in metrically bounded but noncompact subsets which had been known before only in special cases in the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all our theorems guarantee under general logical conditions such strong uniform versions of non uniform existence statements moreover they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result our metatheorems deal with general classes of spaces like metric spaces hyperbolic spaces cat 0 spaces normed linear spaces uniformly convex spaces as well as inner product spaces

a new 3 d nonspurious discontinuous galerkin spectral element time domain dg setd method for maxwell s equations a new discontinuous galerkin spectral element time domain dg setd method for maxwell s equations based on the field variables mathbf e and mathbf b is proposed to analyze three dimensional 3 d transient electromagnetic phenomena compared to the previous setd method based on the field variables mathbf e and mathbf h the mathbf eh scheme in which different orders of interpolation polynomials for electric and magnetic field intensities are required the newly proposed method can eliminate spurious modes using basis functions with the same order interpolation for electric field intensity and magnetic flux density the mathbf eb scheme consequently it can reduce the number of unknowns and computation load domain decomposition for the mathbf eb scheme setd method is completed via the dg method in addition the mathbf eb scheme setd method is extended to the well posed time domain perfectly matched layer pml to truncate the computation domain when solving open region problems the effectiveness and advantages of the new dg setd method are validated by eigenvalue analysis and numerical results

on a finite element method for solving the three dimensional maxwell equations abstract the aim of this paper is to present a method for solving the time domain three dimensional maxwell equations which can be coupled with a particle solver for this purpose maxwell s equations are reformulated as a constrained wave equation system with lagrange multipliers associated to the conditions 0 and e e 0 we approximate both the fields and the lagrange multipliers with a finite element method using a taylor hood element

a concept of convergence in geodesic spaces abstract a cat 0 space is a geodesic space for which each geodesic triangle is at least as thin as its comparison triangle in the euclidean plane a notion of convergence introduced independently several years ago by lim and kuczumow is shown in cat 0 spaces to be very similar to the usual weak convergence in banach spaces in particular many banach space results involving weak convergence have precise analogues in this setting at the same time many questions remain open

the behavior of fixed point free nonexpansive mappings in geodesic spaces abstract dropping the existence of fixed points of a nonexpansive mapping is an interesting and unusual task in metric fixed point theory hyperbolic geometry proved to be very relevant in the study of the behavior of fixed point free nonexpansive mappings in this work we generalize some of the results in that direction in geodesic spaces more precisely we show under which additional assumptions the picard iterative sequence of a mapping defined on a hyperbolic geodesic space tends to a point of the boundary

alternatives to the rayleigh quotient for the quadratic eigenvalue problem we consider the quadratic eigenvalue problem lambda 2 ax lambda bx cx 0 suppose that u is an approximation to an eigenvector x for instance obtained by a subspace method and that we want to determine an approximation to the corresponding eigenvalue lambda the usual approach is to impose the galerkin condition r theta u theta 2 a theta b c u perp u from which it follows that theta must be one of the two solutions to the quadratic equation u au theta 2 u bu theta u cu 0 an unnatural aspect is that if u x the second solution has in general no meaning when u is not very accurate it may not be clear which solution is the best moreover when the discriminant of the equation is small the solutions may be very sensitive to perturbations in u in this paper we therefore examine alternative approximations to lambda we compare the approaches theoretically and by numerical experiments the methods are extended to approximations from subspaces and to the polynomial eigenvalue problem

effective results on compositions of nonexpansive mappings abstract this paper provides uniform bounds on the asymptotic regularity for iterations associated to a finite family of nonexpansive mappings we obtain our quantitative results in the setting of r convex spaces a class of geodesic spaces which generalizes metric spaces with a convex geodesic bicombing

least squares finite element method and preconditioned conjugate gradient solution a least squares variational procedure for first order systems of differential equations and an approximate formulation based on finite elements are developed error estimates a condition number bound and analysis of weighting factors are given steepest descent and conjugate gradient solution procedures are examined and an appropriate preconditioner constructed which is demonstrated to yield rapid convergence and to be insensitive to problem size numerical studies of rates of convergence for a test problem are given

elimination of spurious modes in finite element analysis abstract a common cause of spurious non physical modes that arise in either finite difference or finite element derived eigenvalue problems is identified it is shown that these modes are the result of an excessively flexible system the flexibility is removed by constraining the problem an analogy is drawn between electromagnetic wave propagation and acoustic wave propagation in liquids to show that the spurious modes encountered in both cases are due to fundamentally the same cause results are presented to show that constraining the problem yields a significant reduction in the number of spurious modes a big improvement in the quality of the eigenvector of the physical modes and only a marginal increase in the error in the eigenvalue for the low order modes

the origin of spurious solutions in computational electromagnetics it is commonly believed that the divergence equations in the maxwell equations are redundant for transient and timeharmonic problems therefore most of the numerical methods in computational electromagnetics solve only two first order curl equations or the second order curl curl equations this misconception is the true origin of spurious modes and inaccurate solutions in computational electromagnetics by studying the div curl system this paper clarifies that the first order maxwell equations are not overdetermined and the divergence equations must always be included to maintain the ellipticity of the system in the space domain to guarantee the uniqueness of the solution and the accuracy of the numerical methods and to eliminate the infinitely degenerate eigenvalue this paper shows that the common derivation and usage of the second order curl curl equations are incorrect and that the solution of helmholtz equations needs the divergence condition to be enforced on an associated part of the boundary the div curl method and the least squares method introduced in this paper provide rigorous derivation of the equivalent second order maxwell equations and their boundary conditions the node based least squares finite element method lsfem is recommended for solving the first order full maxwell equations directly examples of the numerical solutions by lsfem are given to demonstrate that the lsfem is free of spurious solutions

approximating nonabelian free groups by groups of homeomorphisms of the real line abstract we show that for a large class c of finitely generated groups of orientation preserving homeomorphisms of the real line the following holds given a group g of rank k in c there is a sequence of k markings g s n n n whose limit in the space of marked groups is the free group of rank k with the standard marking the class we consider consists of groups that admit actions satisfying mild dynamical conditions and a certain self similarity type hypothesis examples include thompson s group f higman thompson groups stein thompson groups various bieri strebel groups the golden ratio thompson group and finitely presented nonamenable groups of piecewise projective homeomorphisms for the case of thompson s group f we provide a new and considerably simpler proof of this fact proved by brin in 4

the dehn function of richard thompson s group f is quadratic we prove that the dehn function that is the smallest isoperimetric function of the r thompson s group f is quadratic

groups of piecewise projective homeomorphisms the group of piecewise projective homeomorphisms of the line provides straightforward torsion free counterexamples to the so called von neumann conjecture the examples are so simple that many additional properties can be established

averaged alternating reflections in geodesic spaces abstract we study the nonexpansivity of reflection mappings in geodesic spaces and apply our findings to the averaged alternating reflection algorithm employed in solving the convex feasibility problem for two sets in a nonlinear context we show that weak convergence results from hilbert spaces find natural counterparts in spaces of constant curvature moreover in this particular setting one obtains strong convergence

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

the bieri neumann strebel invariants via newton polytopes we study the newton polytopes of determinants of square matrices defined over rings of twisted laurent polynomials we prove that such newton polytopes are single polytopes rather than formal differences of two polytopes this result can be seen as analogous to the fact that determinants of matrices over commutative laurent polynomial rings are themselves polynomials rather than rational functions we also exhibit a relationship between the newton polytopes and invertibility of the matrices over novikov rings thus establishing a connection with the invariants of bieri neumann strebel bns via a theorem of sikorav we offer several applications we reprove thurston s theorem on the existence of a polytope controlling the bns invariants of a 3 manifold group we extend this result to free by cyclic groups and the more general descending hnn extensions of free groups we also show that the bns invariants of poincare duality groups of type mathtt f in dimension 3 and groups of deficiency one are determined by a polytope when the groups are assumed to be agrarian that is their integral group rings embed in skew fields the latter result partially confirms a conjecture of friedl we also deduce the vanishing of the newton polytopes associated to elements of the whitehead groups of many groups satisfying the atiyah conjecture we use this to show that the l 2 torsion polytope of friedl luck is invariant under homotopy we prove the vanishing of this polytope in the presence of amenability thus proving a conjecture of friedl luck tillmann

metrics and embeddings of generalizations of thompson s group f the distance from the origin in the word metric for generalizations f p of thompson s group f is quasi isometric to the number of carets in the reduced rooted tree diagrams representing the elements of f p this interpretation of the metric is used to prove that every f p admits a quasiisometric embedding into every f q and also to study the behavior of the shift maps under these embeddings the first example of a finitely presented infinite simple group was discovered by r j thompson in 1965 this group g arose in the study of logic and it has many interesting descriptions in 1974 higman 10 used a description as the group of automorphisms of the jonsson tarski algebra to generalize g to an infinite family of groups g has another description as a group of homeomorphisms of the cantor set it has subgroups which may be described as homeomorphism groups of different objects for instance the circle and the unit interval among the people studying these generalizations are brown 3 and bieri and strebel in a set of unpublished notes 1 in this paper we will concentrate on the groups f p which correspond to the groups fp in 3 these families of groups have also been extensively studied by brin and guzman 2 the group f 2 commonly known as thompson s group f can be regarded as the group of piecewise linear orientation preserving homeomorphisms of the interval 0 1 which have breakpoints only at dyadic points and whose slopes where defined are powers of two in 1984 brown and geoghegan 4 found it to be the first example of a finitely presented infinite dimensional torsion free fp group this fact has been extended to all f p in 3 and studied by stein 11 where generalizations to more general groups of homeomorphisms with general rational breakpoint sets are considered cleary 7 has studied these properties for groups of piecewise linear homeomorphisms with irrational breakpoints and slopes in 5 burillo established the relationship between the word metric of thompson s group f and an estimate of the distance derived from the unique normal form of the elements this algebraic estimate is quasi isometric to the word metric and was used to prove that some subgroups are nondistorted in f in this paper we find a geometric estimate of the word metric in terms of rooted tree diagrams see 6 show that this is quasi isometric to the word metric and use this interpretation to prove that some embeddings of f p into f q including some of those outlined in 2 are quasi isometric the interpretations of these embeddings in terms of received by the editors september 25 1998 and in revised form august 11 1999 2000 mathematics subject classification primary 20f65 secondary 20f05 20f38 20e99 05c25 c 2000 american mathematical society

the schauder fixed point theorem in geodesic spaces abstract we give a direct proof of schauder s fixed point theorem in the setting of geodesic metric spaces generalizing the classical schauder s theorem and improving a recent version of this theorem in cat spaces as an application we prove an existence result for a variational inequality in the setting of cat spaces

electromagnetics via the taylor galerkin finite element method on unstructured grids traditional techniques for computing electromagnetic solutions in the time domain rely on finite differences these so called fdtd finite difference time domain methods are usually defined only on regular lattices of points and can be too restrictive for geometrically demanding problems great geometric flexibility can be achieved by abandoning the regular latticework of sample points and adopting an unstructured grid an unstructured grid allows one to place the grid points anywhere one chooses so that curved boundaries can be fit with ease and local regions in which the field gradients are steep can be selectively resolved with a fine mesh in this paper we present a technique for solving maxwell s equations on an unstructured grid based on the taylor galerkin finite element method we present several numerical examples which reveal the fundamental accuracy and adaptability of the method although our examples are in two dimensions the techniques and results generalize readily to 3d

eb scheme based hybrid se fe dgtd method for multiscale em simulations this communication presents an eb scheme subdomainlevel discontinuous galerkin time domain dgtd method for multiscale simulations it is an extension of the previous subdomain level dgtd research by combining the degree of freedom efficiency of spectral element time domain method and the mesh flexibility of the finite element time domain method thus the multiscale problems can be solved efficiently by separating the geometrically fine and coarse parts and meshing them with hexahedrons and tetrahedrons respectively via a nonconformal mesh the implicit explicit runge kutta method is applied to the eb scheme dgtd method to obtain an efficient time integration approach

automorphisms of generalized thompson groups 0 1 results we study the automorphisms of some generalizations of thompson s groups and their underlying structures the automorphism groups of two of thompson s original groups were analyzed in 2 and were shown to be small and unexotic our results differ sharply from 2 in that we show that the automorphism groups of the generalizations are large and have exotic elements the term exotic is from 1 and is explained later in this introduction richard j thompson introduced the triple of infinite groups f t g in the 1960s and showed that they have several interesting properties these groups were later generalized to infinite groups fn fn tn r gn r for n 2 and r 1 for which f f2 f2 t t2 1 and g g2 1 further background is given in the second part of this introduction the automorphism groups aut f and aut t were analyzed in 2 in this paper we study aut fn aut fn and aut tn n 1 for n 2 we find that all three contain exotic elements and that the complexity increases as n increases we also find that the structures of aut fn and aut tn n 1 are closely related while differing significantly from the structure of aut fn this difference increases as n increases our analysis is only partial in that we discover large subgroups of these groups containing exotic elements but we do not know whether the subgroups 1

a quadratic rate of asymptotic regularity for cat 0 spaces abstract in this paper we obtain a quadratic bound on the rate of asymptotic regularity for the krasnoselski mann iterations of nonexpansive mappings in cat 0 spaces whereas previous results guarantee only exponential bounds the method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of groetsch s theorem obtained by kohlenbach using methods from mathematical logic so called proof mining

mathematical analysis of a finite element method without spurious solutions for computation of dielectric waveguides in this paper we propose a finite element method to compute dielectric waveguides which does not present spurious solutions in addition to a theoretical analysis showing the good stability properties of this method we give numerical results for some classical test problems

the embedding of certain linear and abelian groups in finitely presented simple groups in 2 g higman showed than any countable locally finite group can be embedded in a finitely presented simple group in this paper it is shown that any linear group gl n z over the integers and any countable abelian group whose torsion factor group is a subgroup of an x module oieimi where i is a countable set m is an infinite cyclic module and x is the subset of rationals whose denominators are powers of the integer p can be embedded in a finitely presented simple group i am grateful to j s wilson who pointed out to me that it is not necessary to assume that the torsion free x module is finitely generated both of the above results are proved using the construction procedure described in 3 the first result also shows that some new finitely presented simple groups can be constructed using the procedures in 3 for gl 3 z cannot be embedded in any of the previously known finitely presented simple groups see 2 the definitions techniques and notation of 3 will be freely used in this work it will be assumed that the reader is familiar with the first two sections the defining relations a b c and d and the fourth section of 3 but a knowledge of the rest of 3 is not essential it will be useful to have recalled parts of the basic construction that were only mentioned in 3 but will be used heavily in what follows from a homomorphism 0 from a free group f to a wreath product f wr s we obtain the following diagram see 3

simple groups separated by finiteness properties we show that for every positive integer n there exists a simple group that is of type mathrm f n 1 but not of type mathrm f n for n ge 3 these groups are the first known examples of this kind they also provide infinitely many quasi isometry classes of finitely presented simple groups the only previously known infinite family of such classes due to caprace remy consists of non affine kac moody groups over finite fields our examples arise from rover nekrashevych groups and contain free abelian groups of infinite rank

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

an algebraic classification of some solvable groups of homeomorphisms abstract we produce two separate algebraic descriptions of the isomorphism classes of the solvable subgroups of the group pl o i of piecewise linear orientation preserving homeomorphisms of the unit interval under the operation of composition and also of the generalized r thompson groups f n the first description is as a set of isomorphism classes of groups which is closed under three algebraic operations and the second is as the set of isomorphism classes of subgroups of a countable collection of easily described groups we show the two descriptions are equivalent

a new efficient 3d discontinuous galerkin time domain dgtd method for large and multiscale electromagnetic simulations a new discontinuous galerkin time domain dgtd method for solving the 3d time dependent maxwell s equations via the electric field intensity e and magnetic flux density b fields is proposed for the first time it uses curl conforming and divergence conforming basis functions for e and b respectively with the same order of interpolation in this way higher accuracy is achieved at lower memory consumption than the conventional approach based on the field variables e and h the centered flux and riemann solver are both used to treat interfaces with non conforming meshes and both explicit runge kutta method and implicit crank nicholson method are implemented for time integration numerical examples for realistic cases will be presented to verify that the proposed method is a non spurious and efficient dgtd scheme

asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces abstract we further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their picard iterates we use methods of proof mining to obtain an explicit quantitative version of a generalization to geodesic spaces of a result on the asymptotic behavior of picard iterates for firmly nonexpansive mappings proved by reich and shafrir from this result we obtain effective uniform bounds on the asymptotic regularity for firmly nonexpansive mappings besides this we derive effective rates of asymptotic regularity for sequences generated by two algorithms used in the study of the convex feasibility problem in a nonlinear setting

quasi isometrically embedded subgroups of thompson s groupf abstract the goal of this paper is to study subgroups of thompson s group f which are isomorphic to f z n and f f a result estimating the norm of an element of thompson s group is found and this estimate is used to prove that these particular subgroups are quasi isometrically embedded

asymptotic behavior of resolvents of coaccretive operators in the hilbert ball abstract we study the resolvents of coaccretive operators in the hilbert ball with special emphasis on the asymptotic behavior of their compositions and metric convex combinations we consider the case where the given coaccretive operators share a common fixed point inside the ball as well as the case where they share a common sink point on its boundary we establish weak convergence in the former case and strong convergence in the latter we also present two related convergence results for a continuous implicit scheme

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000

a linear nonconforming finite element method for maxwell s equations in two dimensions part i frequency domain we suggest a linear nonconforming triangular element for maxwell s equations and test it in the context of the vector helmholtz equation the element uses discontinuous normal fields and tangential fields with continuity at the midpoint of the element sides an approximation related to the crouzeix raviart element for stokes the element is stabilized using the jump of the tangential fields giving us a free parameter to decide we give dispersion relations for different stability parameters and give some numerical examples where the results converge quadratically with the mesh size for problems with smooth boundaries the proposed element is free from spurious solutions and for cavity eigenvalue problems the eigenfrequencies that correspond to well resolved eigenmodes are reproduced with the correct multiplicity

topological full groups of one sided shifts of finite type we explore the topological full group g of an essentially principal etale groupoid g on a cantor set when g is minimal we show that g and its certain normal subgroup is a complete invariant for the isomorphism class of the etale groupoid g furthermore when g is either almost finite or purely infinite the commutator subgroup d g is shown to be simple the etale groupoid g arising from a onesided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid for such g g is thought of as a generalization of the higman thompson group we prove that g is of type f and so in particular it is finitely presented this gives us a new infinite family of finitely presented infinite simple groups also the abelianization of g is calculated and described in terms of the homology groups of g

firmly nonexpansive mappings in classes of geodesic spaces firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators in this paper we do a thorough study of fixed point theory and the asymptotic behaviour of picard iterates of these mappings in different classes of geodesic spaces such as uniformly convex w hyperbolic spaces busemann spaces and cat 0 spaces furthermore we apply methods of proof mining to obtain effective rates of asymptotic regularity for the picard iterations msc primary 47h09 47h10 53c22 secondary 03f10 47h05 90c25 52a41