comparative study of methods of various orders for finding repeated roots of nonlinear equations abstract in this paper we are considering 20 families of methods for finding repeated roots of a nonlinear equation the methods are of order up to 8 we use the idea of basin of attraction to compare the methods we found that 4 methods performed best based on 3 quantitative criteria

constructing higher order methods for obtaining the multiple roots of nonlinear equations this paper concentrates on iterative methods for obtaining the multiple roots of nonlinear equations using the computer algebra system mathematica we construct an iterative scheme and discuss the conditions to obtain fourth order methods from it all the presented fourth order methods require one function and two derivative evaluation per iteration and are optimal higher order iterative methods for obtaining multiple roots we present some special methods from the iterative scheme including some known already numerical examples are also given to show their performance

new third order nonlinear solvers for multiple roots abstract two third order methods for finding multiple zeros of nonlinear functions are developed one method is based on chebyshev s third order scheme for simple roots and the other is a family based on a variant of chebyshev s which does not require the second derivative two other more efficient methods of lower order are also given these last two methods are variants of chebyshev s and osada s schemes the informational efficiency of the methods is discussed all these methods require the knowledge of the multiplicity

stochastic renormalization group approach to random point processes a fractal generalization of poisson statistics a generalization of the shlesinger hughes stochastic renormalization method is suggested for random point processes a decimation procedure is introduced in terms of two parameters the probability that a decimation step takes place and the probability that during a decimation step a random dot is removed from the process at each step a random number of dots are removed by this procedure a chain of point processes is generated the renormalized point process is a superposition of the intermediate processes attached to the different steps for a statistical ensemble the fraction lq of systems for which q decimation steps occur is a power function of the fraction q of points which survive q decimation steps lq q 1 df where df 1 ln ln 1 is a fractal exponent smaller than unity df 1 if the random points are initially independent then a complete analysis is possible in this case explicit expressions for the renormalized janossy densities the joint densities and the generating functional of

a third order modification of newton s method for multiple roots in this paper we present a new third order modification of newton s method for multiple roots which is based on existing third order multiple root finding methods numerical examples show that the new method is competitive to other methods for multiple roots

a class of two point sixth order multiple zero finders of modified double newton type and their dynamics under the assumption of the known multiplicity of zeros of nonlinear equations a class of two point sextic order multiple zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double newton type of methods with the introduction of a bivariate weight function dependent on function to function and derivative to derivative ratios higher order convergence is obtained additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases through a variety of test equations numerical experiments strongly support the theory developed in this paper in addition relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction

a branched chain approach to fractal time abstract a branched chain description of fractal time is suggested within its framework the long time tails are the result of a stochastic renormalization procedure the jumps are grouped into blocks of random size the blocks of blocks etc the size of a block being described by a branching chain process the block formation ends up after a random number of steps according to a scale invariant probability law the general scaled form of a multi state waiting time density function is derived as expected this function has a long tail for large times the shlesinger hughes time scaling physica a 109 1981 597 is recovered as a particular case of our approach the fluctuation analysis outlines an essential feature of the theory due to jump clustering the correlation functions have also long tails however unlike the case of true chain reactions the system has the tendency to smooth the fluctuations

a general family of third order method for finding multiple roots abstract in this paper we describe a general family of iterative methods for approximating a multiple root z with multiplicity m of a complex defined function almost of the family of the methods existing in the literature that use two function and one derivative evaluations are a special choice of this general method we give some conditions to have the third order of convergence and we discuss how to choose a small asymptotic error constant which may be affect the speed of the convergence using mathematica with its high precision compatibility we present some numerical examples to confirm the theoretical results

analogies between colored levy noise and random channel approach to disordered kinetics we point out some interesting analogies between colored levy noise and the random channel approach to disordered kinetics these analogies are due to the fact that the probability density of the levy noise source plays a similar role as the probability density of rate coefficients in disordered kinetics although the equations for the two approaches are not identical the analogies can be used for deriving new useful results for both problems the random channel approach makes it possible to generalize the fractional uhlenbeck ornstein processes fuo for space and time dependent colored noise we describe the properties of colored noise in terms of characteristic functionals which are evaluated by using a generalization of huber s approach to complex relaxation phys rev b 31 6070 1985 we start out by investigating the properties of symmetrical white noise and then define the levy colored noise in terms of a langevin equation with a levy white noise source we derive exact analytical expressions for the various characteristic functionals which characterize the noise and a functional fractional fokker planck equation for the probability density functional of the noise at a given moment in time second by making an analogy between the theory of colored noise and the random channel approach to disordered kinetics we derive fractional equations for the evolution of the probability densities of the random rate coefficients in disordered kinetics these equations serve as a basis for developing methods for the evaluation of the statistical properties of the random rate coefficients from experimental data special attention is paid to the analysis of systems for which the observed kinetic curves can be described by linear or nonlinear stretched exponential kinetics

fractals in physics applications and theoretical developments abstract we report about the developments in two subjects in which we have been recently active the first one refers to a reanalysis of the distribution of galaxies and clusters with the methods of modern statistical mechanics contrary to the usual picture of a relatively smooth distribution at large scale we find that fractal and multifractal properties extend up to the observational limits this result should have deep implications also in relation to the recently observed anisotropy of the microwave background radiation the second subject concerns the attempts to develop a theory of fractal growth based on the new concept of the fixed scale transformation we discuss the instructive example of the percolating cluster in which the fst program can be fully implemented we then compare this example with the more complex cases of dla and dbm for which the identification of the scale invariant growth rules is much more complex in this respect the fst points to the key concept that is the scale invariance of screening and noise reduction the state of the art and the main open problems will be critically discussed

a random walk approach to spiral motion the paper deals with random spiral motion described in polar coordinates the trajectory of a particle is described by the distance r from a given point as a function of the angle theta relative to a given direction a stochastic description of the spiral shapes generated by the random variations of r and theta is suggested based on the following assumptions a the ray r is made up of additive contributions corresponding to certain jump processes b the angle phi between two jumps is a random variable selected from a given probability law g phi d phi with finite or infinite moments c the contributions rho 1 rho 2 of the different jumps to the ray r are independent random variables selected from a probability law p rho d rho with finite moments an expression for the generating functional of a random spiral r theta is given in the form of an infinite series which can be used to evaluate the stochastic properties of the ray r the asymptotic properties of the random spiral shapes depend on the function g phi d phi if all moments of the angle between two jumps exist and are finite the average shape is a linear archimedean spiral r theta approximately theta as theta to infinity and the dispersion of the ray increases linearly with the angle delta r2 theta approximately theta as theta to infinity if g phi has a long tail g phi approximately phi 1 h as phi to infinity with 1 h 0 the average shape is a nonlinear fractal archimedean spiral r theta approximately theta h as phi to infinity and the fluctuations of the ray have an intermittent behaviour delta r2 theta approximately theta 2h as phi to infinity a complete analysis is possible in a markovian like case for which the angle between two jumps is exponentially distributed in this case a closed expression tor the generating functional is available and all cumulants of the ray can be computed exactly the mth cumulant which expresses the correlations among the rays at different angles theta 1 theta m is proportional to the minimum angle min theta 1 theta m and the intermittent behaviour is missing an alternative stochastic description is suggested based on the assumption that the number of jumps in a given angle interval is distributed according to an inhomogeneous poisson law this model is also analytically tractable it is less restrictive in the sense that the average shape corresponds to a broader class of spirals including the linear and the fractal archimedean and the logarithmic ones however it cannot be used to describe the intermittent behaviour

on the convergence of an optimal fourth order family of methods and its dynamics in this paper we present the study of the semilocal and local convergence of an optimal fourth order family of methods moreover the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family be means of studying the dynamical behavior parameter spaces are shown and the study of the stability of all the fixed points is presented

fractional oscillator driven by a gaussian noise abstract the time dependent response is studied of a fractional oscillator driven by a gaussian white noise applying the method of characteristic functionals combined with the technique of integral transforms we derive explicit formulas for the variances of position and velocity and the asymptotic power spectrum of the oscillator the effects of the order of fractional derivative on statistical characteristics of the oscillator are analyzed numerically

mass action law versus local contagion dynamics a mean field statistical approach with application to the theory of epidemics a mean field approach for epidemic processes with high migration is suggested by analogy with non equilibrium statistical mechanics for large systems a limit of the thermodynamic type is introduced for which both the total size of the system and the total number of individuals tend to infinity but the population density remains constant in the thermodynamic limit the infection rate is proportional to the product of the proportion of individuals susceptible to infection and the average probability of infection the limit form of the average probability of infection is insensitive to the detailed behaviour of the fluctuations of the number of infectious individuals and may belong to two universality classes 1 if the fluctuation of the number of infectives is non intermittent it increases with the increase of the partial density of infectives and approaches exponentially the asymptotic value one for large densities 2 for intermittent fluctuations obeying a power law scaling the average probability of infection also displays a saturation effect for large densities of infectives but the asymptotic value one is approached according to a power law rather than exponentially for low densities of infectives both expressions for the average probability of infection are linear functions of the proportion of infectives and the infection rate is given by the mass action law

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

generating functional approach to space and time dependent colored noise a stochastic model for time and space dependent colored noise is suggested based on the following assumptions 1 the noise field is generated by a random number of point events corresponding to a correlated point process 2 the contributions of the different events to the noise field are additive each contribution being a random function selected from the same probability density functional an analytical treatment of the noise field described by these assumptions is possible the generating functional of the noise field as well as the corresponding cumulants can be computed exactly all cumulants are explicitly evaluated when the contribution of an event is given by a diffusion equation a detailed analysis of the asymptotic behavior is made for time homogeneous and translationally invariant processes a gaussian random field colored in space and time emerges for very frequent independent events of very small intensities provided that the central moments of the contribution of an event to the noise field are finite and fast decreasing in space the correlations among events lead to corrections of the gauss limit law however the cumulants of the random fields are also finite if the central moments of an event are slowly decreasing in space a different type of asymptotic behavior occurs the cumulants of the noise field become infinite and the resulting field is described by a non gaussian stable law of the l evy type the theory may be applied to the study of stochastic gravitational fluctuations in galactic systems to the analysis of concentration fluctuations for diffusion processes in disordered systems and for the analysis of the influence of environmental fluctuations in continuum mechanics and electrodynamics

the szegő projection sobolev estimates in regular domains the szego projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of diederich fornxess and catlin it preserves local smoothness near boundary points of finite type the bergman projection being linked to the 8 problem plays an important role in several complex variables as demonstrated by recent applications to the boundary behavior of holomorphic maps 7 16 the szego projection is interesting both by analogy and by its connection with the ab problem and hence with unsolvable equations of lewy type while in strictly pseudoconvex domains the bergman and szego projections are both well understood in general much more is known about the bergman projection this is no surprise for the theory of the 8 problem is further advanced than the theory of the 8b problem nonetheless i believe that the szego projection ought not be any more mysterious than the bergman projection this paper is intended to support this contention the most important domains for which the bergman projection is known to be regular are domains with a good a neumann operator here it is shown that in such domains the szego projection also is regular more precisely in smooth bounded pseudoconvex domains satisfying catlin s property p the most general condition presently known that guarantees global regularity of the neumann operator the szego projection is shown to be globally regular theorem 4 1 moreover at boundary points of finite type the szego projection is locally regular theorem 5 1 in both cases the szego projection exactly preserves the degree of differentiability measured in sobolev norms the method of proof is not the obvious one of directly generalizing from the strictly peudoconvex case for there are serious obstacles to such a generalization for instance the approach of analysis on the heisenberg group used by phong and stein 29 to derive estimates for both the bergman and szego projections cannot yet be extended because the theory of singular integrals on weakly pseudoconvex surfaces is still in its early stages 28 15 similarly one does not know how to carry over the fourier integral operator approach of boutet de monvel and sjostrand 10 to the weakly pseudoconvex case explicit construction of cauchy fantappie forms received by the editors march 11 1986 paper presented april 11 1986 to the 826th meeting of the american mathematical society indianapolis special session on several complex variables 1980 mathematics subject classification 1985 revision primary 32a25 32h10 partially supported by national science foundation grants mcs 8201063 and dms 8501758 1987 american mathematical society 0002 9947 87 1 00 25 per page

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

cubic convergence of parameter controlled newton secant method for multiple zeros let f c c have a multiple zero a with integer multiplicity m 1 and be analytic in a sufficiently small neighborhood of a for parameter controlled newton secant method defined by x n 1 x n lf x n 2f x n f x n f x n mf x n f x n n 0 1 2 we investigate the maximal order of convergence and the theoretical asymptotic error constant by seeking the relationship between parameters l and m for various test functions the numerical method has shown a satisfactory result with high precision mathematica programming

on newton type methods for multiple roots with cubic convergence we introduce two families of newton type methods for multiple roots with cubic convergence a further newton type method for multiple roots with cubic convergence is presented that is related to quadrature we also provide numerical tests that show that these new methods are competitive to other known methods for multiple roots

nonequilibrium fluctuation dissipation relations for independent random rate processes with dynamical disorder a class of rate processes with dynamical disorder is investigated based on the two following assumptions a the system is composed of a random number of particles or quasiparticles which decay according to a first order kinetic law b the rate coefficient of the process is a random function of time with known stochastic properties the formalism of characteristic functionals is used for the direct computation of the dynamical averages the suggested approach is more general than the other approaches used in the literature it is not limited to a particular type of stochastic process and can be applied to any type of random evolution of the rate coefficient we derive an infinity of exact fluctuation dissipation relations which establish connections among the moments of the survival function and the moments of the number of surviving particles the analysis of these fluctuation dissipation relations leads to the unexpected result that in the thermodynamic limit the fluctuations of the number of partic

complex dynamics of derivative free methods for nonlinear equations the dynamical behavior of two iterative derivative free schemes steffensen and m4 methods is studied in case of quadratic and cubic polynomials the parameter plane is analyzed for both procedures on quadratic polynomials different dynamical planes are showed when the mentioned methods are applied on particular cubic polynomials with real or complex coefficients the property of immersion of the basins of attraction in all cases is analyzed

comparing the basins of attraction for kanwar bhatia kansal family to the best fourth order method there are relatively few optimal fourth order methods for solving nonlinear algebraic equations having roots of known multiplicity m in a previous paper we have compared 5 such methods two of which require the evaluation of the m 1 st root we have used the basin of attraction idea to recommend the best optimal fourth order method here we compare the family of methods developed by kanwar et al 2013 to the best known method we will also point out some mistake in deriving that family

basins of attraction for several methods to find simple roots of nonlinear equations there are many methods for solving a nonlinear algebraic equation the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss several third and fourth order methods to find simple zeros the relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments the effect of the extraneous roots on the basins is also discussed

new optimal class of higher order methods for multiple roots permitting f xn 0 finding multiple zeros of nonlinear functions pose many difficulties for many of the iterative methods a major difficulty in the application of iterative methods is the selection of initial guess such that neither guess is far from zero nor the derivative is small in the vicinity of the required root otherwise the methods would fail miserably finding a criterion for choosing initial guess is quite cumbersome and therefore more effective globally convergent algorithms for multiple roots are still needed therefore the aim of this paper is to present an improved optimal class of higher order methods having quartic convergence permitting f x 0 in the vicinity of the required root the present approach of deriving this optimal class is based on weight function approach all the methods considered here are found to be more effective and comparable to the similar robust methods available in literature

construction of fourth order optimal families of iterative methods and their dynamics in this paper we propose a general class of fourth order optimal multi point methods without memory for obtaining simple roots this class requires only three functional evaluations viz two evaluations of function f xn f yn and one of its first order derivative f xn per iteration further we show that the well known ostrowski s method and king s family of fourth order procedures are special cases of our proposed schemes one of the new particular subclasses is a biparametric family of iterative methods by using complex dynamics tools its stability is analyzed showing stable members of the family further on one of the parameters is fixed and the stability of the resulting class is studied on the other hand the accuracy and validity of new schemes is tested by a number of numerical examples by comparing them with recent and classical optimal fourth order methods available in the literature it is found that they are very useful in high precision computations

some fourth order nonlinear solvers with closed formulae for multiple roots in this paper we present six new fourth order methods with closed formulae for finding multiple roots of nonlinear equations the first four of them require one function and three derivative evaluation per iteration the last two require one function and two derivative evaluation per iteration several numerical examples are given to show the performance of the presented methods compared with some known methods

extraneous fixed points basin boundaries and chaotic dynamics for schro der and ko nig rational iteration functions the schroder and konig iteration schemes to find the zeros of a polynomial functiong z represent generalizations of newton s method in both schemes iteration functionsf m z are constructed so that sequencesz n 1 f m z n converge locally to a rootz ofg z aso z n z m it is well known that attractive cycles other than the zerosz may exist for newton s method m 2 asm increases the iteration functions add extraneous fixed points and cycles whether attractive or repulsive they affect the julia set basin boundaries the konig functionsk m z appear to minimize such perturbations in the case of two roots e g g z z 2 1 cayley s classical result for the basins of attraction of newton s method is extended for allk m z the existence of chaotic z n sequences is also demonstrated for these iteration methods

on a family of laguerre methods to find multiple roots of nonlinear equations there are several methods for solving a nonlinear algebraic equation having roots of a given multiplicity m here we compare a family of laguerre methods of order three as well as two others of the same order and show that euler cauchy s method is best we discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction

on two dimensional fractional brownian motion and fractional brownian random field as a generalization of one dimensional fractional brownian motion 1dfbm we introduce a class of two dimensional self similar strongly correlated random walks whose variance scales with power law we report analytical results on the statistical size and shape and segment distribution of its trajectory in the limit of large n the relevance of these results to polymer theory is discussed we also study the basic properties of a second generalization of 1dfbm the two dimensional fractional brownian random field 2dfbrf it is shown that the product of two 1dfbms is the only 2dfbrf which satisfies the self similarity defined by sinai

a physical interpretation of age dependent master equations abstract physical systems described by fast and slow relaxing variables are considered the underlying microscopic equations are assumed to be markovian considering that the macroscopic state of the system is characterized by the set of slow relaxing variables we show that the age state probability density obeys a system of age dependent master equations with time and age dependent transition rates if the time scales attached to the slow and fast relaxing variables are well differentiated there is a strong correlation between the age of a fluctuation state and the corresponding number of transition events an alternative approach based on haken s slaving principle is developed for time homogeneous microscopic equations the adiabatic elimination of fast relaxing variables leads to a system of age dependent master equations with time independent and age dependent transition rates these equations are equivalent to a generalized master equation with an inhomogeneous term

basin attractors for various methods for multiple roots abstract there are several methods for approximating the multiple zeros of a nonlinear function when the multiplicity is known the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss all known methods of orders two to four and present the basin of attraction for several examples

a hopf like equation and perturbation theory for differential delay equations we extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations because the phase spaces of differential delay equations are infinite dimensional phase space densities for these systems are functionals we derive a hopf like functional differential equation governing the evolution of these densities the functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory a necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given finally we show that the evolution equation for the density functional is the fourier transform of the infinite dimensional version of the kramers moyal expansion

a cubic order variant of newton s method for finding multiple roots of nonlinear equations abstract a second derivative free iteration method is proposed below for finding a root of a nonlinear equation f x 0 with integer multiplicity m 1 x n 1 x n f x n f x n f x n f x n f x n n 0 1 2 we obtain the cubic order of convergence and the corresponding asymptotic error constant in terms of multiplicity m and parameters and various numerical examples are presented to confirm the validity of the proposed scheme

high order nonlinear solver for multiple roots a method of order four for finding multiple zeros of nonlinear functions is developed the method is based on jarratt s fifth order method for simple roots and it requires one evaluation of the function and three evaluations of the derivative the informational efficiency of the method is the same as previously developed schemes of lower order for the special case of double root we found a family of fourth order methods requiring one less derivative thus this family is more efficient than all others all these methods require the knowledge of the multiplicity

a smoothing property of the bergman projection let b be the bergman projection associated to a domain on which the bar partial neumann operator is compact we show that arbitrary l 2 derivatives of bf are controlled by derivatives of f taken in a single distinguished direction as a consequence functions not contained in c infty overline omega that are mapped by b to c infty overline omega are explicitly described

generating functional approach to multichannel parallel relaxation with application to the problem of direct energy transfer in fractal systems with dynamic disorder a model for multichannel parallel relaxation is suggested based on the following assumptions a an individual channel is characterized by a set of continuous state variables the corresponding relaxation rate is a function of the state variables as well as of the time interval for which the channel is open b the number of channels is a random variable described by a correlated point process defined in the space of state parameters of an individual channel analytical expressions for the generating functional of the overall relaxation rate and for the average survival function are derived in terms of the generating functional of the point process the general formalism is applied to the problem of direct energy transfer from excited donors to acceptors in fractal systems with dynamic disorder it is assumed that the number of acceptors obeys a poissonian distribution law with a constant average density in a df dimensional fractal structure embedded in a ds dimensional euclidean space ds 1 2 3 and th

evolution towards ergodic behavior of stationary fractal random processes with memory application to the study of long range correlations of nucleotide sequences in dna abstract the possible occurrence of ergodic behavior for large times is investigated in the case of stationary random processes with memory it is shown that for finite times the time average of a state function is generally a random variable and thus two types of cumulants can be introduced for the time average and for the statistical ensemble respectively in the limit of infinite time a transition from the random to the deterministic behavior of the time average may occur resulting in an ergodic behavior the conditions of occurrence of this transition are investigated by analyzing the scaling behavior of the cumulants of the time average a general approach for the computation of these cumulants is developed explicit computations are presented both for short and long memory in the particular case of separable stationary processes for which the cumulants of a statistical ensemble can be factorized into products of functions depending on binary time differences in both cases the ergodic behavior emerges for large times provided that the cumulants of a statistical ensemble decrease to zero as the time differences increase to infinity the analysis leads to the surprising conclusion that the scaling behavior of the cumulants of the time average is relatively insensitive to the type of memory considered both for short and long memory the cumulants of the time average obey inverse different from zero for large time differences then the time averaage is random even as the length of the total time interval tends to infinity and the ergodic behavior no longer holds the theory is applied to the study of long range correlations of nucleotide sequences in dna in this case the length t of a sequence of nucleotides plays the role of the time variable a proportionality relationship is established between the cumulants of the pyrimidine excess in a sequence of length t and the cumulants of the time length average of the probability of occurrence of a pyrimidine it is shown that the statistical analysis of the dna data presented in the literature is consistent with the occurrence of the ergodic behavior for large lengths the implications of the approach to the analysis of the large time behavior of stochastic cellular automata and of fractional brownian motion are also investigated

a sixth order family of three point modified newton like multiple root finders and the dynamics behind their extraneous fixed points a class of three point sixth order multiple root finders and the dynamics behind their extraneous fixed points are investigated by extending modified newton like methods with the introduction of the multivariate weight functions in the intermediate steps the multivariate weight functions dependent on function to function ratios play a key role in constructing higher order iterative methods extensive investigation of extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction numerical experiments applied to a number of test equations strongly support the underlying theory pursued in this paper relevant dynamics of the proposed methods is well presented with a variety of illustrative basins of attraction applied to various test polynomials

analogs of renormalization group transformations in random processes abstract we review the properties of a real space renormalization group transformation of the free energy including the existence of oscillatory terms multiplying the non analytic part of the free energy we then construct stochastic processes which incorporate into probability distributions the features of the free energy scaling equation the essential information is obtainable from the scaling equation and a direct solution for a probability is not necessary these random processes are shown to be generated directly from cantor sets in a spatial representation the ensuing random process exhibits a transition between gaussian and fractal behavior in the fractal regime the trajectories will in an average sense form self similar clusters in a temporal representation the random process exhibits a transition between an asymptotically constant renewal rate and fractal behavior the fractal regime represents a frozen state with only transient effects allowed and is related to charge transport in glasses

on levy or stable distributions and the williams watts model of dielectric relaxation this paper is concerned with the levy or stable distribution function defined by the fourier transform q alpha left z right frac 1 2 pi int infty infty exp left izu left u right alpha right du with 0 alpha leqslant 2 when 2 it becomes the gauss distribution function and when 1 the cauchy distribution when 2 the distribution has a long inverse power tail q alpha left z right sim frac gamma left 1 alpha right sin tfrac 1 2 pi alpha pi left z right 1 alpha in the regime of small if logz 1 the distribution is mimicked by a log normal distribution we have derived rapidly converging algorithms for the numerical calculation ofq z for various in the range 0 1 the functionq z appears naturally in the williams watts model of dielectric relaxation in that model one expresses the normalized dielectric parameter as in n left omega right equiv in n left omega right i in n left omega right int 0 infty e i omega t left d phi left t right mathord left vphantom d phi left t right dt right kern nulldelimiterspace dt right dt with phi left t right exp left t mathord left vphantom t tau right kern nulldelimiterspace tau right alpha it has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form of t e n is shown to be directly related toq z it is also shown that if the williams watts exponential is expressed as a weighted average of exponential relaxation functions exp left t mathord left vphantom t tau right kern nulldelimiterspace tau right alpha int 0 infty g left lambda alpha right e lambda t dt the weight functiong is expressible as a stable distribution some suggestions are made about physical models that might lead to the williams watts form of t

spatiotemporal correlation of colored noise we calculated the noise correlation in a gaussian stochastic process that is non ensuremath delta function correlated in both space and time the colored noise obeys a linear reaction diffusion langevin equation with gaussian white noise our result is a generalization of the ornstein uhlenbeck process to take into account finite correlation in space as well as in time for colored noise

comparison of several families of optimal eighth order methods several families of optimal eighth order methods to find simple roots are compared to the best known eighth order method due to wang and liu 2010 we have tried to improve their performance by choosing the free parameters of each family using two different criteria

basin attractors for various methods abstract there are many methods for the solution of a nonlinear algebraic equation the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss several methods of various orders and present the basin of attraction for several examples it can be seen that not all higher order methods were created equal newton s halley s murakami s and neta johnson s methods are consistently better than the others in two of the examples neta s 16th order scheme was also as good

different anomalies in a jarratt family of iterative root finding methods abstract in this paper the behavior of a jarratt family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family be means of studying the dynamical behavior of this fourth order family of methods parameter spaces are shown and the study of the stability of all the fixed points is presented dynamical planes for members with good and bad dynamical behavior are also provided

lp regularity of the bergman projection on domains covered by the polydisc abstract if a bounded domain can be covered by the polydisc through a rational proper holomorphic map then the bergman projection is l p bounded for p in a non trivial interval depending on the ramified rational covering this result can be applied to the symmetrized polydisc and to the hartogs triangle with exponent

a renormalization approach to a class of exponential random processes with application to the bronchial tree abstract a new stochastic renormalization approach for multi step decay phenomena is developed a simple physical interpretation of the renormalization method is suggested it consists in grouping successions of variable numbers of decay events into blocks the law of probability multiplication leads to an exponential random process y yx1 x2 0 where both the exponents x1 x2 and the basis y0 are random variables the physically consistent solution of the model corresponds to a super strong renormalization regime the dependence of the moments of the decay parameter on the number of decay steps q can be exactly determined it consists in a linear superposition of inverse power laws in q modulated by periodic functions in in q having different periods the theory is applied to the renormalization of the bronchial tree our computation shows that the lung structure is very tolerant to fluctuations this result supports the mechanism of morphogenesis of fractal biological organs suggested by west ann biomed eng 18 1990 135 the possibilities of application to the scattering phenomena in one dimensional disordered media are also investigated

on optimal fourth order iterative methods free from second derivative and their dynamics in this paper new fourth order optimal root finding methods for solving nonlinear equations are proposed the classical jarratt s family of fourth order methods are obtained as special cases we then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three the basins of attraction of existing optimal methods and our method are presented and compared to illustrate their performance