uniformization of sierpinski carpets in the plane let si i i be a countable collection of jordan curves in the extended complex plane widehat mathbb c that bound pairwise disjoint closed jordan regions if the jordan curves are uniform quasicircles and are uniformly relatively separated then there exists a quasiconformal map f colon widehat mathbb c rightarrow widehat mathbb c such that f si is a round circle for all i i this implies that every sierpinski carpet in widehat mathbb c whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round sierpinski carpet by a quasisymmetric map

three step fixed point quasi newtonmethods for unconstrained optimisation multistep quasi newton methods were introduced by ford and moghrabi 1 they address the problem of the unconstrained minimisation of a function f n whose gradient and hessian are denoted by g and g respectively these methods generalised the standard construction of quasi newton methods and were based on employing interpolatory polynomials to utilise information from more than one previous step in a series of papers ford and moghrabi 2 5 have developed various techniques for determining the parametrisation of the interpolating curves in 2 they introduced two step metric based methods which determine the set of parameter values required through measuring distances between various pairs of the iterates employed in the current interpolation one of the most successful methods in 2 was found to be in the fixed point class in which the parametrisation of the interpolating curve is determined at each iteration by reference to distances measured from a fixed iterate as suggested in 1 multistep quasi newton methods can be constructed for any number of steps in this paper we therefore extend the previous work by describing the development of some three step methods which use the fixed point approach and data derived from the latest four iterates the experimental results provide evidence that the new methods offer a significant improvement in performance when compared with the standard bfgs method and the unit spaced three step method particularly as the dimension of the test problems grows

new implicit updates in multi step quasi newton methods for unconstrained optimisation multi step quasi newton methods for optimisation using data from more than one previous step to revise the current approximate hessian were introduced by ford and moghrabi in j comput appl math 50 1994 305 where they showed how to construct such methods by means of interpolating curves these methods also utilise standard quasi newton formulae but with the vectors normally employed in the formulae replaced by others determined from a multi step version of the secant equation some methods the accumulative and fixed point approaches for defining the parameter values which correspond to the iterates on the interpolating curve were presented by ford and moghrabi in optim methods software 2 1993 357 both the accumulative and the fixed point methods measure the distances required to parameterise the interpolating polynomials via a norm defined by a positive definite matrix m the fixed point algorithm which takes m to be the current approximate hessian was found experimentally to be the best of the six multi step methods studied in ford and moghrabi 1993 all of which exhibited improved numerical performance by comparison with the standard single step bfgs method to produce a better parameterisation of the interpolation ford comput math appl 42 2001 1083 developed the idea of implicit update methods the fundamental concept here is to determine an improved version of the hessian approximation to be used in computing the metric while avoiding the computational expense of actually calculating the improved version two implicit methods denoted by i2 and i3 were developed from f2 in ford 2001 the method i2 employed parameter values generated from an implicit single step bfgs update while i3 used values from an implicit two step update in this paper we describe the derivation of new implicit updates which are similar to i3 the experimental results we present show that one of the new implicit methods produces markedly better performance than the existing implicit methods particularly as the dimension of the test problem grows

nonlinear conjugate gradient methods with structured secant condition for nonlinear least squares problems in this paper we deal with conjugate gradient methods for solving nonlinear least squares problems several newton like methods have been studied for solving nonlinear least squares problems which include the gauss newton method the levenberg marquardt method and the structured quasi newton methods on the other hand conjugate gradient methods are appealing for general large scale nonlinear optimization problems by combining the structured secant condition and the idea of dai and liao 2001 20 the present paper proposes conjugate gradient methods that make use of the structure of the hessian of the objective function of nonlinear least squares problems the proposed methods are shown to be globally convergent under some assumptions finally some numerical results are given

square sierpinski carpets and lattes maps we prove that every quasisymmetric homeomorphism of a standard square sierpinski carpet s p p ge 3 odd is an isometry this strengthens and completes earlier work by the authors bonk and merenkov in ann math 2 177 591 643 2013 theorem 1 2 we also show that a similar conclusion holds for quasisymmetries of the double of s p across the outer peripheral circle finally as an application of the techniques developed in this paper we prove that no standard square carpet s p is quasisymmetrically equivalent to the julia set of a postcritically finite rational map

the use of alternation and recurrences in two step quasi newton methods of the multistep quasi newton methods introduced by the authors in 1 the mostsuccessful was the so called fixed point method using the existing hessian approximation to compute at each iteration the parameters required in the interpolation in order to avoid the burden of computing the additional matrix vector products required by this approach approximations based on the secant equation were proposed in 2 a different approach to dealing with this difficulty was proposed in which standard single step quasi newton updates were alternated on successive iterations with two step updates so that approximations were no longer necessary recent work has shown that the quantities required to compute the parameters referred to above may be computed exactly by means of a recurrence so that the technique of alternation is no longer the only alternative if we wish to avoid approximations in this paper we describe the derivation of this recurrence we present the results of a range of numerical experiments to compare and evaluate the three approaches of approximation alternation and recurrence finally we show how the use of recurrences may be extended to multistep methods employing three or more steps

two new conjugate gradient methods based on modified secant equations following the approach proposed by dai and liao we introduce two nonlinear conjugate gradient methods for unconstrained optimization problems one of our proposed methods is based on a modified version of the secant equation proposed by zhang deng and chen and zhang and xu and the other is based on the modified bfgs update proposed by yuan an interesting feature of our methods is their account of both the gradient and function values under proper conditions we show that one of the proposed methods is globally convergent for general functions and that the other is globally convergent for uniformly convex functions to enhance the performance of the line search procedure we also propose a new approach for computing the initial steplength to be used for initiating the procedure we provide a comparison of implementations of our methods with the efficient conjugate gradient methods proposed by dai and liao and hestenes and stiefel numerical test results show the efficiency of our proposed methods

on the construction of minimisation methods of quasi newton type abstract the secant equation which underlies all standard quasi newton minimisation methods arises from the use of a linear function to model the gradient along a chosen direction we present new minimisation algorithms derived by replacing this linear model with a more general one involving a free parameter which is determined by using information contained in the current approximate hessian the use of such a model can give more flexibility in the criteria to be satisfied during the line search the new methods can operate as soon as a reasonable approximation to the hessian has been accumulated and may in one sense be viewed as acceleration techniques for quasi newton methods

on the dynamics of a family of singularly perturbed rational maps abstract in this paper we study the dynamical behavior of the family of complex rational maps which is given by f z z n z 2 n n 1 z 2 n 3 n 1 where n 2 and c 2 n 2 1 this family of rational maps can be seen as a perturbation of the unicritical polynomial z z n if is small we prove that the julia set j f of f is either a quasicircle a cantor set of circles a sierpinski carpet or a degenerated sierpinski carpet provided one of the free critical points of f is escaping to the origin or to the infinity in particular we prove that there exists suitable such that the julia set j f is a cantor set of circles but f is not topologically conjugate to any mcmullen map on their corresponding julia sets

area and hausdorff dimension of sierpinski carpet julia sets we prove the existence of rational maps whose julia sets are sierpinski carpets having positive area such rational maps can be constructed such that they either contain a cremer fixed point a siegel disk or are infinitely renormalizable we also construct some sierpinski carpet julia sets with zero area but with hausdorff dimension two moreover for any given number s in 1 2 we prove the existence of sierpinski carpet julia sets having hausdorff dimension exactly s

a quasi newton method employing direct secant updates of matrix factorizations a quasi newton algorithm is presented which employs an initial factorization of the approximate jacobian matrix the method then updates the upper and lower triangular factors directly at each step iterates are generated using forward and backward substitution employing the updated factorizations the method is shown to be locally and q superlinearly convergent and a numerical algorithm is presented

local and superlinear convergence of quasi newton methods based on modified secant conditions for solving unconstrained minimization problems quasi newton methods are popular iterative methods the secant condition which employs only the gradient information is imposed on these methods several researchers paid attention to other secant conditions to get a better approximation of the hessian matrix of the objective function recently zhang et al new quasi newton equation and related methods for unconstrained optimization j optim theory appl 102 1999 147 167 and zhang and xu properties and numerical performance of quasi newton methods with modified quasi newton equations j comput appl math 137 2001 269 278 proposed the modified secant condition which uses both gradient and function value information in order to get a higher order accuracy in approximating the second curvature of the objective function they showed the local and q superlinear convergence property of the bfgs like and dfp like updates based on their proposed secant condition in this paper we incorporate one parameter into this secant condition to smoothly switch the standard secant condition and the secant condition of zhang et al we consider a modified broyden family which includes the bfgs like and the dfp like updates proposed by zhang et al we prove the local and q superlinear convergence of our method

a quadratically convergent krawczyk like algorithm in this paper we introduce a method for computing a solution of a nonlinear system which is similar to that proposed by r krawczyk computing 4 1969 pp 187 201 our method however needs considerably less work per step starting with an interval vector we give a criterion under which the method is convergent to the solution of the system if a solution is contained in the interval vector if the starting vector contains no solution then the method will break down after a finite number of steps

properties and numerical performance of quasi newton methods with modified quasi newton equations quasi newton qn equation plays a core role in contemporary nonlinear optimization the usual qn equation employs only the gradients but ignores the available function value information in this paper we derive a class of modified qn equations with a vector parameter which use both available gradient and function value information the modified quasi newton methods maintain most properties of the usual quasi newton methods meanwhile they achieve a higher order accuracy in approximating the second order curvature of the problem functions than the usual ones do numerical experiments are reported which support the theoretical analyses and show the advantages of the modified qn methods over the usual ones

conjugate gradient methods based on secant conditions that generate descent search directions for unconstrained optimization conjugate gradient methods have been paid attention to because they can be directly applied to large scale unconstrained optimization problems in order to incorporate second order information of the objective function into conjugate gradient methods dai and liao 2001 proposed a conjugate gradient method based on the secant condition however their method does not necessarily generate a descent search direction on the other hand hager and zhang 2005 proposed another conjugate gradient method which always generates a descent search direction in this paper combining dai liao s idea and hager zhang s idea we propose conjugate gradient methods based on secant conditions that generate descent search directions in addition we prove global convergence properties of the proposed methods finally preliminary numerical results are given

minimum curvature multistep quasi newton methods abstract we consider multistep quasi newton methods for unconstrained optimization problems such methods were introduced by the authors 1 3 and employ interpolating polynomials to utilize data from the m most recent iterations for the case m 2 we observe that there is a free parameter used in constructing the interpolation which is essentially at our disposal we propose as a criterion for determining this parameter the minimization of a measure of the curvature of the interpolating polynomial thus producing a smooth interpolant we show how this minimum curvature problem may be solved cheaply and effectively at each iteration the performance of an algorithm which employs this technique is compared numerically with that of the standard bfgs method and of previously introduced multistep methods 2 3

alternating multi step quasi newton methods for unconstrained optimization we consider multi step quasi newton methods for unconstrained optimization these methods were introduced by the authors ford and moghrabi 5 6 8 who showed how an interpolating curve in the variable space could be used to derive an appropriate generalization of the secant equation normally employed in the construction of quasi newton methods one of the most successful of these multi step methods employs the current approximation to the hessian to determine the parametrization of the interpolating curve and hence the derivatives which are required in the generalized updating formula however certain approximations were found to be necessary in the process in order to reduce the level of computation required which must be repeated at each iteration to acceptable levels in this paper we show how a variant of this algorithm which avoids the need for such approximations may be obtained this is accomplished by alternating on successive iterations a single step and a two step method the results of a series of experiments which show that the new algorithm exhibits a clear improvement in numerical performance are reported

on the use of function values in unconstrained optimisation abstract by the use of a nonlinear model for the gradient of the objective function along a chosen direction we show how information available via values of the objective function may be efficiently utilised in an optimisation method of quasi newton type numerical experiments indicate that computational gains are possible by such means

using function values in multi step quasi newton methods in previous work the authors 1993 1994 developed the concept of multi step quasi newton methods based on the use of interpolating polynomials determined by data from the m most recent steps different methods for parametrizing these polynomials were studied by the authors 1993 and several methods were shown empirically to yield substantial gains over the standard one step bfgs method for unconstrained optimization in this paper we will consider the issue of how to incorporate function value information within the framework of such multi step methods this is achieved in the case of two step methods through the use of a carefully chosen rational form to interpolate the three most recent iterates the results of numerical experiments on the new methods are reported

the dai liao nonlinear conjugate gradient method with optimal parameter choices minimizing two different upper bounds of the matrix which generates search directions of the nonlinear conjugate gradient method proposed by dai and liao two modified conjugate gradient methods are proposed under proper conditions it is briefly shown that the methods are globally convergent when the line search fulfills the strong wolfe conditions numerical comparisons between the implementations of the proposed methods and the conjugate gradient methods proposed by hager and zhang and dai and kou are made on a set of unconstrained optimization test problems of the cuter collection the results show the efficiency of the proposed methods in the sense of the performance profile introduced by dolan and more

concerning the convergence of inexact newton methods abstract in this study we use inexact newton methods to find solutions of nonlinear operator equations on banach spaces with a convergence structure our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered banach space in this way the metric properties of the examined problem can be analyzed more precisely moreover this approach allows us to derive from the same theorem on the one hand semi local results of kantorovich type and on the other hand global results based on monotonicity considerations by imposing very general lipschitz like conditions on the operators involved on the one hand we cover a wider range of problems and on the other hand by choosing our operators appropriately we can find sharper error bounds on the distances involved than before furthermore we show that special cases of our results reduce to the corresponding ones already in the literature finally our results are used to solve integral equations that cannot be solved with existing methods

globally convergent modified perry s conjugate gradient method abstract conjugate gradient methods are probably the most famous iterative methods for solving large scale optimization problems in scientific and engineering computation characterized by the simplicity of their iteration and their low memory requirements in this paper we propose a new conjugate gradient method which is based on the mbfgs secant condition by modifying perry s method our proposed method ensures sufficient descent independent of the accuracy of the line search and it is globally convergent under some assumptions numerical experiments are also presented

on the use of curvature estimates in quasi newtonian methods abstract in previous work ford and ghandhari 1989 considered the use of nonlinear gradient models for quasi newton methods by means of such models values of the objective function in addition to values of the gradient were utilised in updating formulae numerical experiments provided evidence that such an approach could yield gains in terms of the evaluations required but at the cost of the solution of a nonlinear equation at each iteration in a subsequent paper ford and ghandhari to appear have developed alternative models which enable the parameter to be determined explicitly in this paper we shall consider further models of the general type introduced in the latter paper which determine the parameter explicitly by employing estimates of the curvature of the objective function thus following a technique first introduced by biggs 1971 1973 we conclude by presenting the results of numerical tests on the new methods

a multi iterate method to solve systems of nonlinear equations we propose an extension of secant methods for nonlinear equations using a population of previous iterates contrarily to classical secant methods where exact interpolation is used we prefer a least squares approach to calibrate the linear model we propose an explicit control of the numerical stability of the method we show that our approach can lead to an update formula in that case we prove the local convergence of the corresponding undamped quasi newton method finally computational comparisons with classical quasi newton methods highlight a significant improvement in terms of robustness and number of function evaluations we also present numerical tests showing the robust behavior of our method in the presence of noise

multi step quasi newton methods for optimization abstract quasi newton methods update at each iteration the existing hessian approximation or its inverse by means of data deriving from the step just completed we show how multi step methods employing in addition data from previous iterations may be constructed by means of interpolating polynomials leading to a generalization of the secant or quasi newton equation the issue of positive definiteness in the hessian approximation is addressed and shown to depend on a generalized version of the condition which is required to hold in the original single step methods the results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach by comparison with single step methods particularly as the dimension of the problem increases

on the solution of highly structured nonlinear equations abstract we introduce a quasi newton update for nonlinear equations which have a jacobian with sparse triangular factors and consider its application through an algorithm of deuflhard to the solution of boundary value problems by multiple shooting

improved hessian approximation with modified secant equations for symmetric rank one method symmetric rank one sr1 is one of the competitive formulas among the quasi newton qn methods in this paper we propose some modified sr1 updates based on the modified secant equations which use both gradient and function information furthermore to avoid the loss of positive definiteness and zero denominators of the new sr1 updates we apply a restart procedure to this update three new algorithms are given to improve the hessian approximation with modified secant equations for the sr1 method numerical results show that the proposed algorithms are very encouraging and the advantage of the proposed algorithms over the standard sr1 and bfgs updates is clearly observed

quasisymmetric rigidity of square sierpinski carpets we prove that every quasisymmetric self homeomorphism of the standard 1 3 sierpinski carpet s3 is a euclidean isometry for carpets in a more general family the standard 1 p sierpinski carpets sp p 3 odd we show that the groups of quasisymmetric self maps are finite dihedral we also establish that sp and sq are quasisymmetrically equivalent only if p q the main tool in the proof for these facts is a new invariant a certain discrete modulus of a path family that is preserved under quasisymmetric

planar relative schottky sets and quasisymmetric maps a relative schottky set in a planar domain is a subset of obtained by removing from open geometric discs whose closures are in and are pairwise disjoint in this paper we study quasisymmetric and related maps between relative schottky sets of measure zero we prove in particular that quasisymmetric maps between such sets in jordan domains are conformal locally bi lipschitz and that their first derivatives are locally lipschitz we also provide a locally bi lipschitz uniformization result for relative schottky sets in jordan domains and establish rigidity with respect to local quasisymmetric maps for relative schottky sets in the unit disc

a new two step gradient type method for large scale unconstrained optimization in this paper we propose some improvements on a new gradient type method for solving large scale unconstrained optimization problems in which we use data from two previous steps to revise the current approximate hessian the new method which we considered resembles to that of barzilai and borwein bb method the innovation features of this approach consist in using approximation of the hessian in diagonal matrix form based on the modified weak secant equation rather than the multiple of the identity matrix in the bb method using this approach we can obtain a higher order accuracy of hessian approximation when compares to other existing bb type method by incorporating a simple monotone strategy the global convergence of the new method is achieved practical insights into the effectiveness of the proposed method are given by numerical comparison with the bb method and its variant

a nonlinear model for function value multistep methods abstract we develop a framework employing scaling functions for the construction of multistep quasi newton methods for unconstrained optimization these methods utilize values of the objective function they are constructed via interpolants of the m 1 most recent iterates gradient evaluations and possess a free parameter which introduces an additional degree of flexibility this permits the interpolating functions to assimilate information in the form of function values which is readily available at each iteration motivated by previous experience 1 with the use of function values in multistep methods we investigate the incorporation of this information in the construction of the hessian approximation at each iteration in an attempt to accelerate convergence we concentrate on a specific example from the general family of methods corresponding to a particular choice of the scaling function and from it derive three new algorithms the relative numerical performance of these methods is assessed and the most successful of them is then compared with the standard bfgs method and with an earlier algorithm utilizing function values also developed by the authors 1

quasisymmetries of sierpinski carpet julia sets abstract we prove that if is a quasisymmetric homeomorphism between sierpinski carpets that are julia sets of postcritically finite rational maps then is the restriction of a mobius transformation this implies that the group of quasisymmetric homeomorphisms of a sierpinski carpet julia set of a postcritically finite rational map is finite

direct secant updates of matrix factorizations this paper presents a new context for using the sparse broyden update method to solve systems of nonlinear equations the setting for this work is that a newton like algorithm is assumed to be available which incorporates a workable strategy for improving poor initial guesses and providing a satisfactory jacobian matrix approximation whenever required the total cost of obtaining each jacobian matrix or the cost of factoring it to solve for the newton step is assumed to be sufficiently high to make it attractive to keep the same jacobian approximation for several steps this paper suggests the extremely convenient and apparently effective technique of applying the sparse broyden update directly to the matrix factors in the iterations between reevaluations in the hope that fewer fresh factorizations will be required the strategy is shown to be locally and q superlinearly convergent and some encouraging numerical results are presented

a new super memory gradient method with curve search rule in this paper we propose a new super memory gradient method with curve search rule for unconstrained optimization problems the method uses previous multi step iterative information and some curve search rules to generate new iterative points at each iteration this makes the new method converge stably and be more suitable for solving large scale optimization problems than other similar methods we analyze the global convergence and convergence rate under some mild conditions numerical experiments show that some new algorithms are available and effective in practical computation

rigidity of schottky sets we call the complement of a union of at least three disjoint round open balls in the unit sphere bbb s n a schottky set we prove that every quasisymmetric homeomorphism of a schottky set of spherical measure zero to another schottky set is the restriction of a m obius transformation on bbb s n in the other direction we show that every schottky set in bbb s 2 of positive measure admits nontrivial quasisymmetric maps to other schottky sets these results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries

a sierpinski carpet with the co hopfian property motivated by questions in geometric group theory we define a quasisymmetric co hopfian property for metric spaces and provide an example of a metric sierpinski carpet with this property as an application we obtain a quasi isometrically co hopfian gromov hyperbolic space with a sierpinski carpet boundary at infinity in addition we give a complete description of the quasisymmetry group of the constructed sierpinski carpet this group is uncountable and coincides with the group of bi lipschitz transformations

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

implicit updates in multistep quasi newton methods abstract we consider multistep quasi newton methods for unconstrained optimization these methods were introduced by ford and moghrabi 1 2 who showed how interpolating curves could be used to derive a generalization of the secant equation the relation normally employed in the construction of quasi newton methods one of the most successful of these multistep methods makes use of the current approximation to the hessian to determine the parametrization of the interpolating curve in the variable space and hence the generalized updating formula in this paper we investigate the use of implicit updates to the approximate hessian in an attempt to determine a better parametrization of the interpolation while avoiding the computational burden of actually carrying out the update and thus improve the numerical performance of such algorithms

a new class of spectral conjugate gradient methods based on a modified secant equation for unconstrained optimization conjugate gradient methods have played a special role for solving large scale optimization problems due to the simplicity of their iteration convergence properties and their low memory requirements in this work we propose a new class of spectral conjugate gradient methods which ensures sufficient descent independent of the accuracy of the line search moreover an attractive property of our proposed methods is that they achieve a high order accuracy in approximating the second order curvature information of the objective function by utilizing the modified secant condition proposed by babaie kafaki et al s babaie kafaki r ghanbari n mahdavi amiri two new conjugate gradient methods based on modified secant equations journal of computational and applied mathematics 234 2010 1374 1386 further a global convergence result for general functions is established provided that the line search satisfies the wolfe conditions our numerical experiments indicate that our proposed methods are preferable and in general superior to the classical conjugate gradient methods in terms of efficiency and robustness