two time space tradeoffs for element distinctness two time space tradeoffs for element distinctness are given the first one shows t2s n3 on a branching program using minimum operations by a result of yao 1982 this implies the same bounds for linear queries the second result extends one by duris and galil 1984 who constructed a boolean function that requires t2s n3 on a k headed turing machine here it is shown that their proof also holds for element distinctness a more natural problem

universal scaling behaviour for iterated maps in the complex plane according to the theory of schroder and siegel certain complex analytic maps possess a family of closed invariant curves in the complex plane we have made a numerical study of these curves by iterating the map and have found that the largest curve is a fractal when the winding number of the map is the golden mean the fractal curve has universal scaling properties and the scaling parameter differs from those found for other types of maps also for this winding number there are universal scaling functions which describe the behaviour asn of theq n th iterates of the map whereq n is thenth fibonacci number

work by robert kalaba on automated sensitivity analysis this paper reviews work by robert kalaba on the development of a program nasa for the automated sensitivity analysis of parametrized nonlinear systems over parameter intervals nasa incorporates a fast and efficient algorithm feed for higher order partial derivative evaluation as well as an adaptive homotopy continuation procedure for obtaining all required initial conditions applications are envisioned for fields such as economics and the biological sciences where models tend to be complex and closed form solutions are difficult to obtain

dynamical analysis on cubic polynomials of damped traub s method for approximating multiple roots abstract in this paper the performance of a parametric family including newton s and traub s schemes on multiple roots is analyzed the local order of convergence on nonlinear equations with multiple roots is studied as well as the dynamical behavior in terms of the damping parameter on cubic polynomials with multiple roots the fixed and critical points and the associated parameter plane are some of the characteristic dynamical features of the family which are obtained in this work from the analysis of these elements we identify members of the family of methods with good numerical properties in terms of stability and efficiency both for finding the simple and multiple roots and also other ones with very unstable behavior

tight lower bounds for st connectivity on the nnjag model directed st connectivity is the problem of deciding whether or not there exists a path from a distinguished node s to a distinguished node t in a directed graph we prove a time space lower bound on the probabilistic nnjag model of poon proc 34th annual symposium on foundations of computer science palo alto ca 1993 pp 218 227 let n be the number of nodes in the input graph and s and t be the space and time used by the nnjag respectively we show that for any delta 0 if an nnjag uses space s in o n 1 delta then t in 2 omega log 2 n s otherwise t in 2 omega log 2 n log n over s log log n times ns log n 1 2 in a preliminary version of this paper by edmonds and poon proc 27th annual acm symposium on theory of computing las vegas nv 1995 pp 147 156 a lower bound of t in 2 omega log 2 n log n over s log log n times ns log n 1 2 was proved our result greatly improves the previous lower bound of st in omega n 2 log n on the jag model by barnes and edmonds proc 34th annual symposium on foundations of computer science palo alto ca 1993 pp 228 237 and that of s 1 3 t in omega n 4 3 on the nnjag model by edmonds time space lower bounds for undirected and directed st connectivity on jag models ph d thesis university of toronto toronto on canada 1993 our lower bound is tight for s in o n 1 delta for any delta 0 matching the upper bound of barnes etal proc 7th annual ieee conference on structure in complexity theory boston ma 1992 pp 27 33 as a corollary of this improved lower bound we obtain the first tight space lower bound of omega log 2 n on the nnjag model no tight space lower bound was previously known even for the more restricted jag model

critical points on the boundaries of siegel disks let be a polynomial map of the riemann sphere of degree at least two we prove that if has a siegel disk g on which the rotation number satisfies a diophantine condition then the boundary of g contains a critical point

choosing the most stable members of kou s family of iterative methods abstract in this manuscript we analyze the dynamical anomalies of a parametric family of iterative schemes designed by kou et al it is known that its order of convergence is three for any arbitrary value of the parameter but it has order four and it is optimal in the sense of kung traub s conjecture when a specific value is selected among all the elements of this family one can choose this fourth order element or any of the infinite members of third order of convergence if only the speed of convergence is considered however the stability of the methods plays an important role in their reliability when they are applied on different problems this is the reason why we analyze in this paper the dynamical behavior on quadratic polynomials of the mentioned family the study of fixed points and their stability joint with the critical points and their associated parameter planes show the richness of the class and allow us to find members of it with excellent numerical properties as well as other ones with very unstable behavior some test functions are analyzed for confirming the theoretical results

a time space tradeoff for element distinctness in a time space tradeoff for sorting on non oblivious machines borodin et al b 81 proved that to sort n elements requires ts n2 where t time and s space on a comparison based branching problem although element distinctness and sorting are equivalent problems on a computation tree the stated tradeoff result does not immediately follow for element distinctness or indeed for any decision problem in this paper we are able to show that ts n3 2 for deciding element distinctness or the sign of a permutation

on the multiplier of a repelling fixed point we estimate the multiplier of a repelling fixed point with rational rotation number for a polynomial in one complex variable we show that a canonical branch of the logarithm of lies in a small disk tangent to the imaginary axis the radius of the disk is determined by the degree of the polynomial the rotation number of the fixed point and the escape rates and external arguments of the critical values

dynamics of a family of third order iterative methods that do not require using second derivatives the purpose of this article is to present results that amount to a description of the conjugacy classes of three third order root finding iterative methods that do not require the use of second derivatives for their formulation for complex polynomials of degrees two three and four for degrees two and three a full description of the conjugacy classes is accomplished in each case by a one parameter family of polynomials this is done in such a way that when one applies one of these three root finding iterative methods to the elements of these parametrized families a family of iterative methods is obtained in such a way that its dynamics represents up to conjugacy the dynamics of the corresponding iterative root finding method applied to any complex polynomial having the same degree for degree four analogous partial results are obtained

on the dynamics of laguerre s iteration z n 1 abstract the laguerre algorithm for computing the zeros of polynomial equations is well known many proofs of the global convergence of this method for the class of real polynomials with all real roots and starting from real initial conditions exist in this article the global behavior of the laguerre iteration is studied when applied to zn 1 for this class of polynomials an almost complete description of the dynamics of laguerre is rendered

diophantine conditions imply critical points on the boundaries of siegel disks of polynomials let f be a polynomial map of the riemann sphere of degree at least two we prove that if f has a siegel disk g on which the rotation number satisfies a diophantine condition then either the boundary b of g contains a critical point or b is a lakes of wada indecomposable continuum with one of the lakes containing a critical point consequently if the boundary b of g has only 2 complementary domains then b contains a critical point we also show without any assumption on the rotation number that each proper nondegenerate subcontinuum of the boundary b of g is tree like and any other bounded complementary domain of b is a preperiodic component of the grand orbit of g finally we establish some conditions under which b contains no periodic point

renormalization group analysis of quasi periodicity in analytic maps analytic maps of the form f z e 2 pi i omega z mathcal o z 2 display quasiperiodicity when satisfies a diophantine condition quasiperiodic motion is confined to a neighborhood of the origin known as a siegel domain the boundary of this domain obeys universal scaling relations in this paper we investigate these scaling relations through a renormalization group analysis and we discuss singularities and asymptotic form of the scaling function

solving nonlinear equations by adaptive homotopy continuation standard homotopy continuation methods for solving systems of nonlinear equations require the continuation parameter to move from 0 to 1 along the real line difficulties can occur however if a point of singularity is encountered during the course of the integration to ameliorate these difficulties this paper proposes extending the continuation parameter to complex values and adaptively computing a continuation path in the complex plane that avoids points giving rise to singularities specifically it is proposed that the continuation parameter move from 0 0i to 1 0i along a spider web grid centered at 1 0i in the complex plane the actual path through the grid is determined step by step in accordance with two objectives short path length and avoidance of singular points a two phase homotopy continuation is used to study the implementation of this idea numerical examples are presented which indicate the effectiveness of the approach

julia sets and complex singularities in hierarchical ising models we study the analytical continuation in the complex plane of free energy of the ising model on diamond like hierarchical lattices it is known 12 13 that the singularities of free energy of this model lie on the julia set of some rational endomorphismf related to the action of the migdal kadanoff renorm group we study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin off we prove that for almost all with respect to the harmonic measure geodesics the complex critical exponent is common and compute it

complete comparative static differential equations equations having form 1 arise as first order conditions in microeconomic constrained optimization models as defining characterizations for general equilibrium and macroeconomic models and as steady state solution characterizations for descriptive and optimal growth models see silberberg 12 hansen 7 cass and shell 3 and brock l if y r r is continuously differentiable over a neighborhood of a point x0 a in r n l for which y xo lx 0 and iy x0 or 0 then the implicit function theorem guarantees the existence of a continuously differentiable function x n cr r over some open neighborhood n cro of cx in r such that dx z ix yx x x a lyjx a x g e n cp 2

are there critical points on the boundaries of singular domains generalizing a result of e ghys we prove a general theorem that implies that if a rational functionf of the riemann sphere of degree 2 leaves invariant a singular domainc a disk or a ring on which the rotation number off satisfies a diophantine condition provided that on bar c f is injective then each boundary component ofc contains critical point off the injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off z z n a withne n 2 andae we also show that the singular disks associated to periodic elliptic points off z e az that satisfy a diophantine condition are unbounded in in the end of the paper we give a survey of the theory of iteration of entire functions of

time space trade off lower bounds for randomized computation of decision problems we prove the first time space lower bound trade offs for randomized computation of decision problems the bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs our techniques are extension of those used by ajtai and by beame jayram and saks that applied to deterministic branching programs our results also give a quantitative improvement over the previous results previous time space trade off results for decision problems can be divided naturally into results for functions with boolean domain that is each input variable is 0 1 valued and the case of large domain where each input variable takes on values from a set whose size grows with the number of variables in the case of boolean domain ajtai exhibited an explicit class of functions and proved that any deterministic boolean branching program or ram using space s o n requires superlinear time t to compute them the functional form of the superlinear bound is not given in his paper but optimizing the parameters in his arguments gives t n log log n log log log n for s o n1 e for the same functions considered by ajtai we prove a time space trade off for randomized branching programs with error of the form t n s log n s log log n s in particular for space o n1 e this improves the lower bound on time to ns log n log log n in the large domain case we prove lower bounds of the form t ns log n s log log n s for randomized computation of the element distinctness function and lower bounds of the form t n log n s for randomized computation of ajtai s hamming closeness problem and of certain functions associated with quadratic forms over large fields

on the iteration of a rational function computer experiments with newton s method using newton s method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function this article describes the behavior of newton iteration for cubic polynomials after a change of variables these polynomials can be parametrized by a single complex parameter and the newton transformation has a single critical point other than its fixed points at the roots of the polynomial we describe the behavior of the orbit of the free critical point as the parameter is varied the julia set points where newton s method fail to converge is also pictured these sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle

a stable family with high order of convergence for solving nonlinear equations recently li et al 2014 have published a new family of iterative methods without memory with order of convergence five or six which are not optimal in the sense of kung and traub s conjecture therefore we attempt to modify this suggested family in such a way that it becomes optimal to this end we consider the same two first steps of the mentioned family and furthermore we introduce a better approximation for f z in the third step based on interpolation idea as opposed to the taylor s series used in the work of li et al theoretical dynamical and numerical aspects of the new family are described and investigated in details

moments of balanced measures on julia sets by a theorem of s demko there exists a balanced measure on the julia set of an arbitrary nonlinear rational transformation on the riemann sphere it is proved here that if the transformation admits an attractive or indifferent cycle then there is a point with respect to which all the moments of a balanced measure exist moreover these moments can be calculated exactly an explicit balanced measure is exhibited in an example where the julia set is the whole sphere and for which the moments with respect to any point do not all exist 1 introduction the julia set for a nonlinear rational transformation on the riemann sphere is the closure of the set of all repulsive cycles of the transformation br fa j the location or distribution of the set is of interest in the theory of iterated maps and it may be characterized by an associated invariant measure in this paper we extend previous results by showing that when there is an attractive or indifferent cycle certain moments of such a measure exist and can be calculated exactly from the rational transformation this means for example that when the julia set is contained strictly in the real line all of the orthogonal polynomials on the set can be calculated as can the entries of the corresponding infinite dimensional jacobi matrices the latter are of the type which have recently been shown to furnish model schrodinger equations for crystals with almost periodic potentials bbm this leads us to the following speculation which provides one reason for our interest in the present topic we note first the frequent occurrence in the physical universe of objects steady state systems which are fractals from some point of

a finite algorithm for the exact evaluation of higher order partial derivatives of functions of many variables abstract a formal algorithm is given for the systematic exact evaluation of higher order partial derivatives of functions of many variables the algorithm improves upon wengert s method in two key respects applications are envisioned wherever gradients jacobians hessians and power series expansions could be employed

a time space tradeoff for sorting on a general sequential model of computation in a general sequential model of computation no restrictions are placed on the way in which the computation may proceed except that parallel operations are not allowed we show that in such an unrestricted environment text time cdot text space omega n 2 log n in order to sort n integers each in the range 1 n 2

on the invariant sets of a family of quadratic maps the julia setb for the mappingz z 2 is considered where is a complex parameter for 2 a new upper bound for the hausdorff dimension is given and the monic polynomials orthogonal with respect to the equilibrium measure onb are introduced a method for calculating all of the polynomials is provided and certain identities which obtain among coefficients of the three term recurrence relations are given a unifying theme is the relationship betweenb and chains which is explored for 1 4 2 and for with 1 4 with the aid of the bottcher equation thenb is shown to be a holder continuous curve for 1 4

time space tradeoffs for set operations abstract this paper considers time space tradeoffs for various set operations denoting the time requirement of an algorithm by t and its space requirement by s it is shown that ts n 2 for set complementation and ts n 3 2 for set intersection in the r way branching program model in the more restricted model of comparison branching programs the paper provides two additional types of results a tradeoff of ts n 2 e n derived from yao s lower bound for element distinctness is shown for set disjointness set union and set intersection where e n o log n 1 2 a bound of ts n 3 2 is shown for deciding set equality and set inclusion finally a classification of set operations is presented and it is shown that all problems of a large naturally arising class are as hard as the problems bounded in this paper

nonlocal automated sensitivity analysis previously a complete ordinary differential equation ode system was developed for tracking the solution x of a parameterized system of nonlinear equations 0 x over an interval 0 this paper develops a two phase complex homotopy continuation method for obtaining the required initial conditions at t an initial short artificial continuation is followed by a continuation which essentially proceeds through the physically meaningful function which can ameliorate the problem of artificially induced singularities also the path in the complex plane followed by the continuation parameter in each phase evolves sequentially in an attempt to keep the path both short minimal number of integration steps and numerically stable avoidance of singular points a fortran program nasa has been developed for solving the complete ode system starting with a two phase initialization the program incorporates a fast and efficient procedure for the automatic evaluation of partial derivatives numerical experiments are reported which illustrate the program s effectiveness in ref 1 an analytically complete system of ordinary differential equations odes was developed for finding a solution vector x for a parameterized system of nonlinear equations 0 p x the well known ode dx d x was supplemented with odes for the adjoint a and the determinant 6 of the jacobian matrix vx x this differential system in principle permits nonlocal sensitivity analysis i e the tracking of the solution vector x and the sensitivity vector dx d together with the adjoint a and the determinant i over any closed interval 0 where the determinant i remains nonzero a fortran program was used successfully in ref 1 to solve this differential system for several illustrative examples subsequently the program was modified to incorporate a fast and efficient algorithm feed for the automatic evaluation of higher order partial derivatives 2 nevertheless to make the program a truly practical tool a simple and reliable procedure was still needed for obtaining the initial conditions x a and c required by the differential system at the initial parameter point ct this paper develops a simple but effective homotopy continuation method for obtaining these initial conditions which differs in two principal respects from standard homotopy continuation methods first the proposed continuation essentially proceeds through the function rather than through an artificial construct letting f x d x t a standard continuation method for finding a solu tion vector x x to the system of equations 0 f x would generally involve solving a system of equations of the form

building blocks for quadratic julia sets we obtain results on the structure of the julia set of a quadratic polynomial p with an irrationally indifferent fixed point zo in the iterative dynamics of p in the cremer point case under the assumption that the julia set is a decomposable continuum we obtain a building block structure theorem for the corresponding julia set j j p there exists a nowhere dense subcontinuum b c j such that p b b b is the union of the impressions of a minimally invariant cantor set a of external rays b contains the critical point and b contains both the cremer point zo and its preimage in the siegel disk case under the assumption that no impression of an external ray contains the boundary of the siegel disk we obtain a similar result in this case b contains the boundary of the siegel disk properly if the critical point is not in the boundary and b contains no periodic points in both cases the julia set j is the closure of a skeleton s which is the increasing union of countably many copies of the building block b joined along preimages of copies of a critical continuum c containing the critical point in addition we prove that if p is any polynomial of degree d 2 with a siegel disk which contains no critical point on its boundary then the julia set j p is not locally connected we also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected julia set have homeomorphic julia sets

local and nonlocal comparative static analysis of economic systems a complete differential equation system has been developed for the local and nonlocal comparative static analysis of general parametrized economic systems j x a 0 the differential equations allow both qualitative and quantitative sensitivity studies at parameter points or over parameter intervals a fortran program comstat is now available for the numerical integration of the differential equations the only input requirements are j and initial conditions initialization is rendered routine by a two stage application of comstat derivatives are automatically evaluated via a generalized wengert method freeing the user from ever having to form analytical derivatives comstat is applied to a problem in optimal economic growth to show the sensitivity of the steady state capital labor ratio to changes in the time preference parameter

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique