locally discontinuous but globally continuous galerkin methods for elliptic problems abstract we propose and analyze a stabilized hybrid finite element method for elliptic problems consisting of locally discontinuous galerkin problems in the primal variable coupled to a globally continuous problem in the multiplier numerical analysis shows that the proposed formulation preserves the main properties of the associate dg method such as consistency stability boundedness and optimal rates of convergence in the energy norm and in the l 2 norm for adjoint consistent formulations for using an element based data structure it has basically the same complexity and computational cost of classical conforming finite element methods convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here with accuracy equivalent or even better than the corresponding dg approximations

local connectivity of julia sets for unicritical polynomials we prove that the julia set j f of at most finitely renormalizable unicritical polynomial f z d c with all periodic points repelling is locally connected for d 2 it was proved by yoccoz around 1990 it follows from a priori bounds in a modified principal nest of puzzle pieces the proof of a priori bounds makes use of new analytic tools developed in kl09 that give control of moduli of annuli under maps of high degree

monotonicity of entropy for real multimodal maps in 16 milnor posed the monotonicity conjecture that the set of parameters within a family of real multimodal polynomial interval maps for which the topological entropy is constant is connected this conjecture was proved for quadratic by milnor thurston 17 and for cubic maps by milnor tresser see 18 and also 5 in this paper we will prove the general case

on the hyperbolicity of the period doubling fixed point we give a new proof of the hyperbolicity of the fixed point for the period doubling renormalization operator using the local dynamics near a semi attractive fixed point in a banach space and the theory of holomorphic motions we also give a new proof of the exponential contraction of the feigenbaum renormalization operator in the hybrid class of the period doubling fixed point our proof uses the non existence of invariant line fields in the period doubling tower c mcmullen the topological convergence d sullivan and a new infinitesimal argument

regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

a discontinuous hp finite element method for diffusion problems 1 d analysis abstract this paper presents the mathematical analysis of a new variant of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems not requiring auxiliary variables such as those used in mixed methods the focus of this study is on a class of linear second order boundary value problems for which we prove stability and a priori error estimates in both the finite and infinite dimensional spaces

the combinatorial rigidity conjecture is false for cubic polynomials we show that there exist two cubic polynomials with connected julia sets which are combinatorially equivalent but not topologically conjugate on their julia sets this disproves a conjecture by mcmullen from 1995

consistent discontinuous finite elements in elastodynamics finite element discontinuities with respect to time have recently been extremely used in elastodynamic problems due to their natural utilization in combination with adaptive methods and their efficiency in discontinuity capturing techniques for non smooth problems in this work we present some theoretical aspects and numerical results concerning the use of spatial discontinuities in a consistent finite element method for the same class of problems we first review some formulations for the elastostatic problem and prove two korn like inequalities which are very useful for the derivation of convergence rates in sobolev norms next we present formulations for the dynamic case along with comments on their properties and estimates of convergence rates for smooth solutions followed by numerical investigations of a typically non smooth problem involving classical and emerging variational formulations we also show some numerical experiments with finite element spaces enriched by discontinuous functions other than piecewise lagrangian polynomials

the fibonacci unimodal map the fibonacci recurrence of the critical orbit appeared in the work of branner and hubbard on complex cubic polynomials bh 12 and in yoccoz s work yl y2 on quadratic ones as the worst pattern of recurrence on the other hand a real quadratic fibonacci map was suggested by hofbauer and keller hk as a possible candidate for a map having a wild attractor that is a set a which is the w limit set for lebesgue almost every orbit but is strictly smaller than the w limit set for a generic orbit the w limit set of the critical orbit in hk possesses all known topological properties of wild attractors compare bl2 in fact we will see below that the quadratic fibonacci map does not have a wild attractor however the corresponding question for a map with a degenerate critical point remains open actually the first indication of the fibonacci map appeared in the numerical work of tsuda t related to the belousova zhabotinskii reaction and also in numerical work of shibayama sh more precisely they studied the sequence of fibonacci bifurcations creating the fibonacci map this paper will study topological geometrical and measure theoretical properties of the real fibonacci map our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by sullivan s it turns out that the situation can be understood completely and is of quite regular nature in particular any fibonacci map with negative schwarzian and nondegenerate critical point has an absolutely continuous invariant measure so we deal with a regular type of chaotic dynamics it turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the feigenbaum map its hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter branner and hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits their fibonacci tableau is a basic example which corresponds to one particularly close and regular pattern of recurrence if a complex quadratic map z 1 z2 c realizes this fibonacci tableau then the

clustered generalized finite element methods for mesh unrefinement non matching and invalid meshes in spite of significant advancements in automatic mesh generation during the past decade the construction of quality finite element discretizations on complex three dimensional domains is still a difficult and time demanding task in this paper the partition of unity framework used in the generalized finite element method gfem is exploited to create a very robust and flexible method capable of using meshes that are unacceptable for the finite element method while retaining its accuracy and computational efficiency this is accomplished not by changing the mesh but instead by clustering groups of nodes and elements the clusters define a modified finite element partition of unity that is constant over part of the clusters this so called clustered partition of unity is then enriched to the desired order using the framework of the gfem the proposed generalized finite element method can correctly and efficiently deal with i elements with negative jacobian ii excessively fine meshes created by automatic mesh generators iii meshes consisting of several sub domains with non matching interfaces under such relaxed requirements for an acceptable mesh and for correctly defined geometries today s automated tetrahedral mesh generators can practically guarantee successful volume meshing that can be entirely hidden from the user a detailed technical discussion of the proposed generalized finite element method with clustering along with numerical experiments and some implementation details are presented copyright 2006 john wiley sons ltd

computing elliptic curve discrete logarithms with the negation map it is clear that the negation map can be used to speed up the computation of elliptic curve discrete logarithms with the pollard rho method however the random walks defined on elliptic curve points equivalence class p used by pollard rho will always get trapped in fruitless cycles we propose an efficient alternative approach to resolve fruitless cycles besides the theoretical analysis we also examine the performance of the new algorithm in experiments with elliptic curve groups the experiment results show that we can achieve the speedup by a factor extremely close to 2 using the new algorithm with the negation map which is the best performance reported in the literature

families of rational maps and iterative root finding algorithms in this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root finding algorithms we answer a question of smale s by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more we settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics and we give a classification of all such algorithms in the context of conformal dynamics our main result is the following a stable algebraic family of rational maps is either trivial all its members are conjugate by mobius transformations or affine its members are obtained as quotients of iterated addition on a family of complex tori our classification of generally convergent algorithms follows easily from this result as another consequence of rigidity we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices this implies that bounded analytic functions nearly separate points on the moduli space of a rational map

discontinuous subgrid formulations for transport problems in this paper we develop two discontinuous galerkin formulations within the framework of the two scale subgrid method for solving advection diffusion reaction equations we reformulate using broken spaces the nonlinear subgrid scale nsgs finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh here two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges the amount of subgrid viscosity is scaled by the resolved scale solution at the element level yielding a self adaptive method so that no additional stabilization parameter is required numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous galerkin methods

slyrb measures natural invariant measures for chaotic systems in many applications it is useful to consider not only the set that constitutes an attractor but also if it exists the asymptotic distribution of a typical trajectory converging to the attractor indeed in the physics literature such a distribution is often assumed to exist when it exists it is called a natural invariant measure the results by lasota and yorke and by sinai ruelle and bowen represent two approaches both of which establish the existence of an invariant measure the goal of this paper is to relate the lasota yorke measure for chaotic attractors in one dimensional maps and the sinai ruelle bowen measure for chaotic attractors in higher dimensional dynamical systems we introduce the notion of slyrb measure we pronounce the term slyrb as a single word slurb the srb concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically similarly the slyrb concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin the latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes perhaps winding up distributed throughout a particular lake

the collet eckmann condition for rational functions on the riemann sphere we show that the set of collet eckmann maps has positive lebesgue measure in the space of rational maps on the riemann sphere for any fixed degree d 2

continuous discontinuous galerkin methods stabilized through transfer functions applied to the incompressible elasticity and to the stokes problem abstract in this work we present a family of new continuous discontinuous mixed formulations with interpolation of equal order for velocity and pressure for the stokes problem and incompressible elasticity the degrees of freedom associated to the discontinuous component are eliminated and an approach with computational effort similar to that of the continuous formulation is obtained an analysis of continuity and weak coercivity infsup condition is performed and it is proved that the formulation is continuous and satisfies the infsup condition on an appropriated norm error estimates derived from theorems and the lemma presented are obtained on a similar norm to the one of the gls method numerical experiments with the cavity problem and a smooth solution problem show the robustness of the new formulation

factorization of the eighth fermat number we describe a monte carlo factorization algorithm which was used to factorize the fermat number f8 2256 1 previously f8 was known to be composite but its factors were unknown comments only the abstract is given here the full paper appeared as 2 for a succinct proof of primality of the larger factor q8 of f8 see 3 at the time of this paper lenstra s elliptic curve method ecm had not been invented thus a modification of pollard s rho method 1 was used to factor f8 and several mersenne numbers 4 for the factorization of a larger fermat number by ecm see 5 the smaller factor of f8 is 1238926361552897 the epigram i am now entirely persuaded to employ the method a handy trick on gigantic composite numbers may appeal to readers who wish to memorize this factor

a discontinuous finite element formulation for helmholtz equation abstract a discontinuous finite element formulation for helmholtz problems is presented with c 0 continuity across inter element boundaries enforced in a weak sense depending on two parameters and whose choice is crucial for the performance of the proposed method these parameters are chosen through one dimension numerical experiments aiming at minimizing the pollution error optimal values of these parameters are then adopted in more general situations the accuracy and stability of the proposed formulation for linear and bilinear shape functions are demonstrated in several numerical examples in one and two dimensions

the complexity of finding cycles in periodic functions given a function f over a finite domain d and an arbitrary starting point x the sequence f 0 x f 1 x f 2 x cdots is ultimately periodic such sequences are typically the output of random number generators the cycle problem is to determine the first repeated element f n x in the sequence previous algorithms for this problem have required 3n o 1 operations in this paper we show that n 1 theta 1 sqrt m steps are both necessary and sufficient if m memory cells are available to store values of the function we explicitly consider the performance of the algorithm as a function of the amount of memory available and the relative cost of evaluating f and comparing sequence elements for equality

stability and bifurcations for dissipative polynomial automorphisms of mathbb c 2 we study stability and bifurcations in holomorphic families of polynomial automorphisms of mathbb c 2 we say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there we first show that this defines a meaningful notion of stability which parallels in many ways the classical notion of j stability in one dimensional dynamics define the bifurcation locus to be the complement of the weak stability locus in the second part of the paper we prove that under an assumption of moderate dissipativity the parameters displaying homoclinic tangencies are dense in the bifurcation locus this confirms one of palis conjectures in the complex setting the proof relies on the formalism of semi parabolic bifurcation and the construction of critical points in semi parabolic basins which makes use of the classical denjoy carleman ahlfors and wiman theorems

on the lebesgue measure of the feigenbaum julia set we show that the julia set of the feigenbaum polynomial has hausdorff dimension less than 2 and consequently it has zero lebesgue measure this solves a long standing open question

discontinuous galerkin framework for adaptive solution of parabolic problems non conforming meshes are frequently employed in adaptive analyses and simulations of multi component systems we develop a discontinuous galerkin formulation for the discretization of parabolic problems that weakly enforces continuity across non conforming mesh interfaces a benefit of the dg scheme is that it does not introduce constraint equations and their resulting lagrange multiplier fields as done in mixed and mortar methods the salient features of the formulation are highlighted through an a priori analysis when coupled with a mesh refinement scheme the dg formulation is able to accommodate multiple hanging nodes per element edge and leads to an effective adaptive framework for the analysis of interface evolution problems we demonstrate our approach by analysing the stefan problem of solidification

discontinuous finite element formulations applied to cracked elastic domains the solutions of boundary value problems defined on cracked domains are usually non smooth in the surroundings of the crack in this work we formulate the elasticity problem of a body with such geometric characteristic in a number of equivalent variational alternatives and show that we can take advantage of the theory of discontinuous finite elements in order to approximate its solution in an interesting way at little higher programming cost in comparison with the classical galerkin method the idea consists in splitting the global domain into a number of regions in which local mesh refinements are undertaken independently producing irregular meshes with non matching elements that are suitable to be used in discontinuous finite element methods this strategy seems to be attractive to be employed in situations that we know in advance where the critical regions of the domain are located as well as in adaptive techniques

renormalization horseshoe for critical circle maps we prove that the invariant horseshoe for the renormalization of critical circle maps is globally uniformly hyperbolic with one dimensional unstable direction this extends our earlier result on hyperbolicity of periodic orbits of the renormalization and completes the proof of lanford s program

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

random walks supported on random points ofz nz this paper considers random walks on the integers modn supported onk points and asks how long does it take for these walks to get close to uniformly distributed ifk is a constant greenhalgh showed that at least some constant timesn 2 k 1 steps are necessary to make the distance of the random walk from the uniform distribution small here we show that ifn is prime some constant timesn 2 k 1 steps suffice to make this distance small for almost all choices ofk points the proof uses the upper bound lemma of diaconis and shahshahani and some averaging techniques this paper also explores some cases wherek varies withn in particular ifk logn a we find different kinds of results for different values ofa and these results disprove a conjecture of aldous and diaconis

from invariant curves to strange attractors we prove that simple mechanical systems when subjected to external periodic forcing can exhibit a surprisingly rich array of dynamical behaviors as parameters are varied in particular the existence of global strange attractors with fully stochastic properties is proved for a class of second order odes

on random walks for pollard s rho method we consider pollard s rho method for discrete logarithm computation usually in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated we show that this assumption does not hold for the walk originally suggested by pollard its performance is worse than in the random case we study alternative walks that can be efficiently applied to compute discrete logarithms we introduce a class of walks that lead to the same performance as expected in the random case we show that this holds for arbitrarily large prime group orders thus making pollard s rho method for prime group orders about 20 faster than before

combinatorial rigidity for unicritical polynomials we prove that any unicritical polynomial fc z zd c which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid this implies that the connectedness locus the multibrot set is locally connected at the corresponding parameter values and generalizes yoccoz s theorem for quadratics to the higher degree case

the parallelized pollard kangaroo method in real quadratic function fields we show how to use the parallelized kangaroo method for computing invariants in real quadratic function fields specifically we show how to apply the kangaroo method to the infrastructure in these fields we also show how to speed up the computation by using heuristics on the distribution of the divisor class number and by using the relatively inexpensive baby steps in the real quadratic model of a hyperelliptic function field furthermore we provide examples for regulators and class numbers of hyperelliptic function fields of genus 3 that are larger than those ever reported before

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

rigidity and non local connectivity of julia sets of some quadratic polynomials for an infinitely renormalizable quadratic map f c z mapsto z 2 c with the sequence of renormalization periods km and rotation numbers tm pm qm we prove that if limsup k m 1 log p m 0 then the mandelbrot set is locally connected at c we prove also that if limsup t m 1 1 q m 1 and qm then the julia set of fc is not locally connected and the mandelbrot set is locally connected at c provided that all the renormalizations are non primitive satellite this quantifies a construction of a douady and j hubbard and weakens a condition proposed by j milnor

on local connectivity for the julia set of rational maps newton s famous example we show that newton s cubic methods famous rational maps have a locally connected julia set except in some very specific cases in particular when those maps are infinitely renormalizable their julia set is locally connected and contains small copies of nonlocally connected quadratic julia sets this also holds when newton s method is renormalizable and has cremer points unlike the polynomial case after a dynamical description we show the necessity of the brjuno condition within this family

a discontinuous finite element based domain decomposition method in this work we introduce a somewhat unconventional finite element based nonoverlapping domain decomposition method the convenient space decomposition method csd for parallel computers that can be applied to a wide class of boundary value problems and which can only be implemented efficiently exclusively with primal variables by means of the use of discontinuous elements interesting aspects and properties are pointed out along with proofs for the most relevant results numerical experiments for purely diffusion problems are also presented in order to evaluate the effectiveness of the proposed scheme and to confirm the validity of the related theoretical results

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

puzzle geometry and rigidity the fibonacci cycle is hyperbolic we describe a new and robust method to prove rigidity results in complex dynamics the new ingredient is the geometry of the critical puzzle pieces under control of geometry and complex bounds two generalized polynomial like maps which admit a topological conjugacy quasiconformal outside the filled in julia set are indeed quasiconformally conjugate the proof uses a new abstract removability type result for quasiconformal maps following ideas of heinonen and koskela and of kallunki and koskela optimized for applications in complex dynamics we prove as the first application of this new method that for even criticalities distinct from two the period two cycle of the fibonacci renormalization operator is hyperbolic with dimensional unstable manifold

density of hyperbolicity for classes of real transcendental entire functions and circle maps we prove density of hyperbolicity in spaces of i real transcendental entire functions bounded on the real line whose singular set is finite and real and ii transcendental self maps of the punctured plane which preserve the circle and whose singular set apart from zero and infinity is contained in the circle in particular we prove density of hyperbolicity in the famous arnol d family of circle maps and its generalizations and solve a number of other open problems for these functions including three conjectures by de melo salom ao and vargas we also prove density of real hyperbolicity for certain families as in i but without the boundedness condition our results apply in particular when the functions in question have only finitely many critical points and asymptotic singularities or when there are no asymptotic values and the degree of critical points is uniformly bounded

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

mating non renormalizable quadratic polynomials in this paper we prove the existence and uniqueness of matings of the basilica with any quadratic polynomial which lies outside of the 1 2 limb of mathcal m is non renormalizable and does not have any non repelling periodic orbits

continuous discontinuous galerkin methods applied to elasticity problems abstract a new family of coupled continuous discontinuous galerkin formulations is presented and analyzed in this paper these formulations have some distinguishing properties support to all boundary conditions without differentiating whether the condition is dirichlet or not the continuous part of the formulations can use the discontinuous part to have better accuracy and robustness properties and the discontinuous part has the same stabilized properties of common discontinuous galerkin methods a new promising stability parameter is introduced and its effects analyzed in the numerical experiments some important results related to the stability of the formulations associated with the polynomial degree adopted for the continuous component are also shown

perturbing misiurewicz parameters in the exponential family in one dimensional real and complex dynamics a map whose post singular or post critical set is bounded and uniformly repelling is often called a misiurewicz map in results hitherto perturbing a misiurewicz map is likely to give a non hyperbolic map as per jakobson s theorem for unimodal interval maps this is despite genericity of hyperbolic parameters at least in the interval setting we show the contrary holds in the complex exponential family z mapsto lambda exp z misiurewicz maps are lebesgue density points for hyperbolic parameters as a by product we also show that lyapunov exponents almost never exist for exponential misiurewicz maps the lower lyapunov exponent is almost everywhere the upper lyapunov exponent is non negative and depends on the choice of metric

the boundary of chaos for interval mappings a goal in the study of dynamics on the interval is to understand the transition to positive topological entropy there is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations we prove this conjecture in natural families of smooth interval maps and use it to study the structure of the boundary of mappings with positive entropy in particular we show that in families of mappings with a fixed number of critical points the boundary is locally connected and for analytic mappings that it is a cellular set

a space efficient algorithm for group structure computation we present a new algorithm for computing the structure of a finite abelian group which has to store only a fixed small number of group elements independent of the group order we estimate the computational complexity by counting the group operations such as multiplications and equality checks under some plausible assumptions we prove that the expected run time is o n with n denoting the group order and we explicitly determine the o constants we implemented our algorithm for ideal class groups of imaginary quadratic orders and present experimental results

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

on the dimensions of cantor julia sets of rational maps abstract in this paper we study the dimensions associated with the cantor julia set of a rational map whose fatou set is an attracting domain we prove that if the julia set of such a map contains no persistently recurrent critical points then the conformal dimension and the hausdorff dimension of the julia set are equal

analysis and applications of a generalized finite element method with global local enrichment functions abstract this paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method the procedure combines classical global local finite element method concepts with the partition of unity approach it involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization the local solutions are in turn used to enrich the global space using the partition of unity framework the computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently the effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper we also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution key aspects of the computational implementation in particular the numerical integration of generalized fem approximations built with global local enrichment functions are presented the method is applied to fracture mechanics problems with multiple cracks in the domain our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global local enrichment functions

dynamics of mcmullen maps abstract in this article we develop the yoccoz puzzle technique to study a family of rational maps termed mcmullen maps we show that the boundary of the immediate basin of infinity is always a jordan curve if it is connected this gives a positive answer to the question of devaney higher regularity of this boundary is obtained in almost all cases we show that the boundary is a quasi circle if it contains neither a parabolic point nor a recurrent critical point for the whole julia set we show that the mcmullen maps have locally connected julia sets except in some special cases