higher order efficient class of chebyshev halley type methods construction of two point sixth order methods for simple root is an ambitious and challenging task in numerical analysis therefore the main aim of this paper is to introduce a new highly efficient two point sixth order class of chebyshev halley type methods free from second order derivative for the first time each member of the proposed class requires only four functional evaluations viz two evaluations of function f and two of first order derivative f per full iteration a variety of concrete numerical examples illustrate that our proposed methods are more efficient and perform better than existing two point three point sixth order methods available in the literature from their dynamical study it has been observed that our proposed methods have better stability and robustness as compared to the other existing methods

a note on some recent methods for solving nonlinear equations abstract in this note we present some comments on the recent results concerning iterative methods for solving nonlinear equations in the first part we compare a new method n ujevic a method for solving nonlinear equations appl math comput 174 2006 1416 1426 with the newton and ostrowski method including an extensive analysis of numerical results together with the choice of initial approximations and the computational efficiency of the considered methods in the second part we give a list of recently derived root finding methods and we study the question of priority namely we find out that all reviewed methods were already derived at the beginning of the 60s of the last century one of them dates from the 18th century some remarks on the comparison analysis and the application of iteration methods are included

on new exponential quadratically convergent iterative formulae a class of exponential iterative formulae having quadratic convergence is discussed in this paper the theoretical analysis and numerical experiments show that new exponential iterative formulae are effective and comparable to well known method of newton

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

synchronization of boolean networks with time delays abstract this paper studies analytically synchronization of two deterministic boolean networks with time delays which are unidirectionally coupled in the drive response configuration necessary and sufficient conditions of synchronization are established based on the algebraic representations of logical dynamics in terms of the semi tensor product of matrices the synchronization conditions show that for drive response boolean networks different relations between the inherent state delay and the unidirectional coupling delay may result in distinct synchronization phenomena and both kinds of time delays that ensure synchronization if existent are not unique examples are worked out to illustrate the present results

reversible boolean networks i distribution of cycle lengths abstract we consider a class of models describing the dynamics of n boolean variables where the time evolution of each depends on the values of k of the other variables previous work has considered models with dissipative dynamics here we consider time reversible models which necessarily have the property that every possible point in the state space is an element of one and only one cycle the orbits can be classified by their behavior under time reversal the orbits that transform into themselves under time reversal have properties quite different from those that do not in particular a significant fraction of latter type orbits have lengths enormously longer than orbits that are time reversal symmetric for large k and moderate n the vast majority of points in the state space are on one of the time reversal singlet orbits and a random hopping model gives an accurate description of orbit lengths however for any finite k the random hopping approximation fails qualitatively when n is large enough n 22k as in the dissipative case when k is large typical orbit lengths grow exponentially with n whereas for small enough k typical orbit lengths grow much more slowly with n the numerical data are consistent with the existence of a phase transition at which the average orbit length grows as a power of n at a value of k between 1 4 and 1 7 however in the reversible models the interplay between the discrete symmetry and quenched randomness can lead to enormous fluctuations of orbit lengths and other interesting features that are unique to the reversible case

relevant elements magnetization and dynamical properties in kauffman networks a numerical study abstract this is the first of two papers about the structure of kauffman networks in this paper we define the relevant elements of random networks of automata following previous work by flyvbjerg j phys a 21 1988 l955 l960 and flyvbjerg and kjaer j phys a 21 1988 1695 1718 and we study numerically their probability distributions in the chaotic phase and on the critical line of the model a simple approximate argument predicts that their number scales as n on the critical line while it is linear with n in the chaotic phase and independent on system size in the frozen phase this argument is confirmed by numerical results the study of the relevant elements gives useful information about the properties of the attractors in critical networks where the pictures coming from either approximate computation methods or from simulations are not very clear

the nature of attractors in an asymmetric spin glass with deterministic dynamics the authors study the attractors in an infinite range ising spin glass model with deterministic dynamics where the interactions have asymmetry specified by a parameter k they find a duality relation between the attractors for models with asymmetry parameters k and 1 k the attractors are fixed points or limit cycles of short length except for k 1 at which the average cycle length diverges reminiscent of a phase transition and the model has many similarities to the random map model as well as to the infinite range symmetric spin glass in thermal equilibrium including the fact that a few attractors dominate the weight the extent of this dominance varies from sample to sample and so is given by a non trivial probability distribution pi y which they compute numerically

a one parameter family of quadratically convergent iteration formulae a new one parameter family of iteration methods for solving nonlinear equations is presented the convergence analysis shows that their order of convergence is two numerical examples are given to illustrate the efficiency and performance of the proposed methods

emergent properties in random complex automata abstract studies of large randomly assembled binary boolean node automata have demonstrated that such systems can spontaneously exhibit enormously ordered dynamical behavior an important approach to characterizing these behaviors has consisted of studying ensembles of automata ensemble specifications have been based on 1 choice of boolean function regulating each node in the automaton 2 random or biased mappings of the 2 n automata states into themselves 3 the numbers of inputs k per node this article briefly reviews these approaches reports new results on the expected behavior of threshold automata and automata with a particular class of biased mappings in addition it discusses emergent highly ordered dynamical behavior in random automata rich in a specific class of canalizing boolean functions due to the crystallization of powerful subautomata called forcing structures the ordered behaviors may be of importance in biological evolution in physics and in the design of adaptive automata

dyson s ranks and appell lerch sums denote by p n the number of partitions of n and by n a m n the number of partitions of n with rank congruent to a modulo m we find and prove a general formula for dyson s ranks by considering the deviation of the ranks from the average begin aligned d a m sum n 0 infty left n a m n frac p n m right q n end aligned using appell lerch sum properties we decompose d a m into modular and mock modular parts so that the mock modular component is supported on certain arithmetic progressions whose modulus we can control using our decomposition we show how our formula gives as a straightforward consequence atkin and swinnerton dyer s results on ranks as well as celebrated work of bringmann ono and rhoades s on maass forms we also give an example of how our methods apply to work of lovejoy and osburn on rank differences

the strange man in random networks of automata abstract we have performed computer simulations of kauffman s automata on several graphs such as the regular square lattice and invasion percolation clusters in order to investigate phase transitions radial distributions of the mean total damage dynamical exponent and propagation speeds of the damage when one adds a damaging agent nicknamed strange man despite the increase in the damaging efficiency we have not observed any appreciable change of the transition threshold to chaos neither for the short range nor for the small world case on the square lattices when the strange man is added in comparison to when small initial damages are inserted in the system particularly we have checked the damage spreading when some connections are removed on the square lattice and when one considers special invasion percolation clusters high boundary saturation clusters it is seen that the propagation speed in these systems is quite sensible to the degree of dilution on the square lattice and to the degree of saturation on invasion percolation clusters

on a class of quadratic convergence iteration formulae without derivatives a class of iterative formulae without derivatives are proposed in this paper these methods are shown to be quadratic convergence some numerical tests are given

a family of optimal sixteenth order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth step weighting function a new family of four step optimal multipoint iterative methods of order sixteen for solving nonlinear equations are developed along with their convergence properties numerical experiments with comparison to some existing methods are demonstrated to support the underlying theory

on the construction of iterative methods with at least cubic convergence using the iteration formulas of order two for solving nonlinear equations we present a basic tool for deriving new higher order iterative methods that do not require the computation of the second order or higher order derivatives the presented convergence analysis shows that the order of convergence of the obtained iterative methods are three or higher the comparison with other methods is given

third order and fourth order iterative methods for finding multiple and distinct zeros of non linear equations in this paper we point out and analyse six two step iterative methods for finding multiple as well as distinct zeros of non linear equations we prove that the methods for multiple zeros have third order convergence where as the methods for distinct zeros have fourth order convergence the methods calculate the multiple as well as distinct zeros with high accuracy the numerical tests show their better performance in case of algebraic as well as non algebraic equations

mock modularity of the m d rank of overpartitions abstract we investigate the modular properties of a new partition rank the m d rank of overpartitions in fact this is an infinite family of ranks indexed by the positive integer d that gives both the dyson rank of overpartitions and the overpartition m 2 rank as special cases the m d rank of overpartitions is the holomorphic part of a certain harmonic maass form of weight 1 2 we give the exact transformation of this harmonic maass form along with a few identities for the m d rank

ramanujan s lost notebook vi the mock theta conjectures 1 introduction in this paper we shall consider only ramanujan s two families of fifth order mock theta functions these functions were briefly described in ramanujan s last letter to g h hardy 11 pp 354 3551 and g n watson subsequently 12 131 proved all the assertions about these functions contained in the letter subsequently the identities watson proved were greatly generalized in 2 also the fifth order mock theta functions were shown to have double series expansions involving indefinite quadratic forms 7 however there remains a profound mystery about these functions namely no one including ramanujan has ever proved that these functions are indeed mock theta functions and not just some clever combination of theta functions this problem was described in detail by watson 12 p 2741 and redescribed in 7 pp 113 l 141 in s p 97 eqs 3 5 and 3 6 a formula from the lost notebook was pointed out that would if valid lead to the establishment of at least some of the fifth order mock theta functions as truly mock theta functions in the sense of watson 12 p 78 footnote there are indeed ten such identities live for each of the two families we have two objects in this paper first we wish to show that the identities in each family are equivalent i e if one is true all five are true and if one is false all five are false to introduce our second objective we recall some basic notions from partition theory 4 p 142 8 p 841 the rank of a partition is the largest

tenth order mock theta functions in ramanujan s lost notebook ramanujan s lost notebook contains many results on mock theta functions in particular the lost notebook contains eight identities for tenth order mock theta functions previously the author proved the first six of ramanujan s tenth order mock theta function identities it is the purpose of this paper to prove the seventh and eighth identities of ramanujan s tenth order mock theta function identities which are expressed by mock theta functions and a definite integral l j mordell s transformation formula for the definite integral plays a key role in the proofs of these identities also the properties of modular forms are used for the proofs of theta function identities

a note on some new iterative methods with third order convergence abstract in j h chen some new iterative methods with three order convergence appl math comput 181 2006 1519 1522 some new iterative methods were proposed and proved to be third order convergence in this paper some mistakes are pointed out in the derivation of these new methods an analysis is given to show that these new iterative methods converge only linearly

distribution of local magnetisations in random networks of automata the authors derive an integral equation which gives the distribution p m of local magnetisations for a random network of automata kauffman model they show how from this distribution one can recover the distances between configurations and the fraction of spins in the stable core i e the spins which do not depend on the initial conditions a numerical solution of the integral equation shows that p m has singularities at m 0 or 1 2 and or 1

certain improvements of newton s method with fourth order convergence in this paper we present two new schemes one is third order and the other is fourth order these are improvements of second order methods for solving nonlinear equations and are based on the method of undetermined coefficients we show that the fourth order method is more efficient than the fifth order method due to kou et al j kou y li x wang some modifications of newton s method with fifth order covergence j comput appl math 209 2007 146 152 numerical examples are given to support that the methods thus obtained can compete with other iterative methods

closing probabilities in the kauffman model an annealed computation abstract we use a probabilistic approach to compute the distribution of periods transients and weights of attraction basins in kauffman networks the results for these quantities are obtained in the framework of the annealed approximation first introduced by derrida and pomeau numerical results for the average periods are in good agreement with the computed values of the exponents we report also on some interesting features which cannot be explained within the annealed approximation

relaxation closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks attractors in asymmetric neural networks with deterministic parallel dynamics present a chaotic regime at symmetry where the average length of the cycles increases exponentially with system size and an oscillatory regime at high symmetry where the average length of the cycles is 2 we show both with analytic arguments and numerically that there is a sharp transition at a critical symmetry between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size and the weights of their attraction basins are distributed as in a random map with reversal symmetry the timescale after which cycles are reached grows exponentially with system size n and the exponent vanishes in the symmetric limit where the transition can be related to the dynamics of the infinite system where cycles are never reached using the closing probabilities as a tool we also study the relaxation of the function where is the local field experienced by the neuron i in the symmetric system it plays the role of a ljapunov function which drives the system towards its minima through steepest descent this interpretation survives even if only on the average also for small asymmetry this acts like an effective temperature the larger the asymmetry the faster the relaxation and the higher the asymptotic value reached e reaches very deep minima in the fixed points of the dynamics which are reached with vanishing probability and attains a larger value on the typical attractors which are cycles of length 2

dyson s ranks and maass forms which were defined by ramanujan and watson decades ago in his last letter to hardy dated january 1920 see pages 127 131 of 27 ramanujan lists 17 such functions and he gives 2 more in his lost notebook 27 in his paper the final problem an account of the mock theta functions 32 watson defines 3 further functions surprisingly much remains unknown about these enigmatic series ramanujan s claims about their analytic properties remain open and there is even debate concerning the rigorous definition of such a function despite these seemingly problematic issues ramanujan s mock theta functions indeed possess many striking properties and they have been the subject of an astonishing number of important works for example see 5 6 7 8 12 13 14 18 19 20 23 27 28 32 33 35 36 to name a few watson predicted this high level of activity in his 1936 presidential address to the london mathematical society with his prophetic words see page 80 of 32

hecke type double sums appell lerch sums and mock theta functions i by introducing a dual notion between partial theta functions and appell lerch sums we find and prove a formula which expresses hecke type double sums in terms of appell lerch sums not only does our formula prove classical hecke type sum identities such as those in work of kac and peterson on affine lie algebras and hecke modular forms but once we have the hecke type forms for ramanujan s mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities indeed we obtain a new proof of the mock theta conjectures our formula also applies to positive level string functions associated with admissible representations of the affine lie algebra a 1 1 of kac and wakimoto

some third order families of iterative methods for solving nonlinear equations abstract in this paper we present a convenient third order iteration formula improving newton s method for solving nonlinear equations with any two iteration functions of order two the formula provides one parameter third order family of methods each family member is free from second derivatives and per iteration it requires three evaluations of the given function and its derivative several numerical examples are given to illustrate the efficiency and performance of some of the presented methods

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

three step iterative methods for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations we show that this new iterative method has third order convergence several numerical examples are given to illustrate the efficiency and performance of the new method

third order derivative free sph iterative method for solving nonlinear systems the majority of iterative methods to find roots of a function requires the evaluation of derivatives of this function in this paper based on the basic principle of the sph method s kernel approximation a kernel approximation was constructed to compute first and second order derivatives through taylor series expansion derivatives in our proposed method were replaced in a newton like iterative method to obtain a derivative free sph iterative method for solving nonlinear systems to illustrate that the new method has the same order of convergence as the considered iterative method some numerical examples are presented

on the seventh order mock theta functions in a recent paper h we proved the mock theta conjectures these are identities stated by ramanujan in his lost notebook jr2 pp 19 20 involving two of the 5th order mock 0 functions in fact ramanujan gave one such identity for each of the ten 5th order functions as shown in a g1 these ten identities are all equivalent to the two proved in i hi in this paper we prove the corresponding identities for ramanujan s 7th order mock 0 functions and show how they are related to rank differences of partitions to state our results we first introduce some standard notation if q and x are complex numbers with ql 1 and n is a nonnegative integer then

a new family of secant like method with super linear convergence a new family of secant like method is proposed in this paper further it has been proved that this family has super linear convergence and efficiency is demonstrated by numerical experiments

sixth order mock theta functions two new mock theta functions of the sixth order are defined the main theorem in this paper theorem 1 1 provides four transformation formulas relating the new mock theta functions with ramanujan s mock theta functions of the sixth order two further representations of the new mock theta functions are established lastly a hitherto unproved entry from ramanujan s lost notebook related to sixth order mock theta functions is proved

new high order convergence iteration methods without employing derivatives for solving nonlinear equations abstract a family of new iteration methods without employing derivatives is proposed in this paper we have proved that these new methods are quadratic convergence their efficiency is demonstrated by numerical experiments the numerical experiments show that our algorithms are comparable to well known methods of newton and steffensen furthermore combining the new method with bisection method we construct another new high order iteration method with nice asymptotic convergence properties of the diameters b n a n

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

the modular structure of kauffman networks this is the second paper of a series of two about the structural properties that influence the asymptotic dynamics of random boolean networks here we study the functionally independent clusters in which the relevant elements introduced and studied in our first paper 3 are subdivided we show that the phase transition in random boolean networks can also be described as a percolation transition the statistical properties of the clusters of relevant elements that we call modules give an insight on the scaling behavior of the attractors of the critical networks that according to kauffman have a biological analogy as a model of genetic regulatory systems

new modified regula falsi method for nonlinear equations abstract this paper employs two new iterative methods accelerating convergence after using the classical regula falsi methods such that both the sequence of diameters b n a n n 1 and the sequence of iteration x n x n 1 produced by the new methods quadratically converge to zero the numerical experiments show that new methods are effective and comparable to some well known methods

from random walks to spin glasses abstract the talk was a short review on systems which exhibit non self averaging effects sums of random variables when the distribution has a long tail mean field spin glasses random map models and returns of a random walk to the origin non self averaging effects are identical in the case of sums of random variables and in the spin glass problem as predicted by the replica approach also we will see that for the random map models or for the problem of the returns of a random walk to the origin the non self averaging effects coincide with the results of the replica approach when the number n of replica n 1 2 or n 1

dynamical phase transition in a neural network model with noise an exact solution the dynamical organization in the presence of noise of a boolean neural network with random connections is analyzed for low levels of noise the system reaches a stationary state in which the majority of its elements acquire the same value it is shown that under very general conditions there exists a critical value gc of the noise below which the network remains organized and above which it behaves randomly the existence and nature of the phase transition are computed analytically showing that the critical exponent is 1 2 the dependence of gc on the parameters of the network is obtained these results are then compared with two numerical realizations of the network

some new iterative methods with three order convergence some new iterative formulae having three order convergence are discussed in this paper the theoretical analysis and numerical experiments show that new iterative formulae are effective and comparable to well known method of newton and stefensen method

a new continuation newton like method and its deformation this paper presents a family of new continuation newton like methods and its deformations for computing approximate solutions of the non linear algebraic equation f x 0 from a practical point of view the new methods are vast superior

attractors in fully asymmetric neural networks the statistical properties of the length of the cycles and of the weights of the attraction basins in fully asymmetric neural networks i e with completely uncorrelated synapses are computed in the framework of the annealed approximation which we previously introduced for the study of kauffman networks our results show that this model behaves essentially as a random map possessing a reversal symmetry comparison with numerical results suggests that the approximation could become exact in the large size limit

fourth order convergent iterative method for nonlinear equation in this paper we suggest and analyze a new iterative method for finding approximate solution of the nonlinear equation f x 0 it is shown that proposed method has fourth order convergence several numerical examples are given to illustrate that the method developed in this paper give better results than the other methods including newton method

on some third order iterative methods for solving nonlinear equations using the iteration formulas of second order mamta v kanwar v k kukreja sukhjit singh on a class of quadratically convergent iteration formulae appl math comput 2004 in press for solving single variable nonlinear equations two classes of third order multipoint methods without using second derivative are derived the main advantage of these classes is that they do not fail if the derivative of the function is either zero or very small in the vicinity of the required root further a new family of secant like method with guaranteed super linear convergence is obtained by discrete modifications and their comparison with respect to the existing classical methods is given

multivalley structure in kauffman s model analogy with spin glasses kauffman s model describes a random network of automata the authors calculations indicate that the multivalley structure of the basins of attraction in kauffman s model is very similar to that of infinite range spin glasses the similarity with spin glasses is tested quantitatively by computing the probability that two initial configurations fall into the same valley

some higher order modifications of newton s method for solving nonlinear equations in this paper we consider constructing some higher order modifications of newton s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units this construction can be applied to any iteration formula and per iteration the resulting methods add only one additional function evaluation to increase the order some illustrative examples are provided and several numerical results are given to show the performance of the presented methods

classification of networks of automata by dynamical mean field theory dynamical mean field theory is used to classify the 65536 different networks of binary automata on a square lattice with nearest neighbour interactions application of mean field theory gives 700 different mean field classes which fall in seven classes of different asymptotic dynamics characterised by fixed points and 2 cycles

some quadrature based three step iterative methods for non linear equations in this paper we present three step quadrature based iterative methods for solving non linear equations the convergence analysis of the methods is discussed it is established that the new methods have convergence order six seven and eight respectively numerical tests show that the new methods are comparable with the well known existing methods and in many cases give better results our results can be considered as an improvement and refinement of the previously known results in the literature

statistical properties of randomly broken objects and of multivalley structures in disordered systems the statistical properties of the multivalley structure of disordered systems and of randomly broken objects have many features in common for all these problems if w denotes the weight of the sth piece we show that the probability distributions pi w of the largest piece w p2 w of the second largest piece and n y of y 1 wi always have singularities at w l n w2 l n and y ijn n 1 2 3

simply constructed family of a ostrowski s method with optimal order of convergence in this paper we propose a simple modification over chun s method for constructing iterative methods with at least cubic convergence 5 using iteration formulas of order two we now obtain several new interesting families of cubically or quartically convergent iterative methods the fourth order family of ostrowski s method is the main finding of the present work per iteration this family of ostrowski s method requires two evaluations of the function and one evaluation of its first order derivative therefore the efficiency index of this ostrowski s family is e 43 1 587 which is better than those of most third order iterative methods e 33 1 442 and newton s method e 2 1 414 the performance of ostrowski s family is compared with its closest competitors namely ostrowski s method jarratt s method and king s family in a series of numerical experiments

on a class of quadratically convergent iteration formulae in this paper a new class of iterative formulae having quadratic convergence is presented furthermore these algorithms are comparable to the well known method of newton and the computed results support this theory

cellular automata model for gene networks in order to study the overall behavior of gene networks we propose a simple cellular automata ca model in which each binary gene is connected to k other inputs including itself interacting through asymmetric short and long range couplings using numerical simulations and mean field calculations collective dynamical properties of this ca model were investigated it is shown that the ca exhibits three different dynamical regimes a frozen a marginal and a chaotic phase where an initial damage vanishes remains limited and grows to a finite fraction of the lattice sites respectively the results presented are also consistent with the observed biological scaling laws for the number of differentiated cells and cell length cycles as a function of the number of genes in an organism s1063 651x 97 03303 5

cellular automata and statistical mechanical models we elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata transient and stationary properties of probabilistic automata are linked to surface and bulk properties respectively of restricted statistical mechanical systems it is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata some exact results are derived including duality transformations exact mappings disorder and linear solutions many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties this approach is considered to be a first step toward the design of fully parallel probabilistic systems whose computational abilities rely on the cooperative behavior of their components

eulerian series as modular forms in 1988 hickerson proved the celebrated mock theta conjectures a collection of ten identities from ramanujan s lost notebook which express certain modular forms as linear combinations of mock theta functions in the context of maass forms these identities arise from the peculiar phenomenon that two different harmonic maass forms may have the same non holomorphic parts using this perspective we construct several infinite families of modular forms which are differences of mock theta functions

fractal dimensions in three dimensional kauffman cellular automata the kauffman random networks of automata are studied on a simple cubic lattice by computer simulations each automaton follows random rules depending on its six neighbours and fixed in time a transition between the frozen and the chaotic phase is observed and the fractal dimension of the asymptotic actual damage at the phase transition is measured

a class of exponential quadratically convergent iterative formulae for unconstrained optimization abstract for solving nonlinear univariate and unconstrained optimization problems newton method is an important and basic method which converges quadratically this paper presents a class of exponential iterative formulae convergence analyses show that the proposed methods converge quadratically the efficiencies of the methods are analyzed in terms of the most popular and widely used criterion in comparison with the classical newton method using five test functions numerical results indicate that one of the new exponential iterative formulae is effective and comparable to well known newton s method

exact solution of kauffman s model with connectivity one kauffman s model is a randomly assembled network of boolean automata each automaton receives inputs from at most k other automata its state at discrete time t 1 is determined by a randomly chosen but fixed boolean function of the k inputs at time t the resulting quenched random dynamics of the network demonstrates two phases a frozen and a chaotic phase the authors give an exact solution of the model for connectivity k 1 valid everywhere in the frozen phase and at a critical point valid for finite as well as for infinite networks they discuss the network s critical behaviour and finite size effects the results for the frozen phase presented complement recent exact results for the chaotic phase obtained for k infinity

on iterative methods for nonlinear equations several iterative methods have been suggested and analyzed for solving nonlinear equation f x 0 all these methods can be classified as one step and two step methods in this paper we have suggested and analyzed several two step methods for solving nonlinear equations we have also shown that these two step methods are quadratically convergent several numerical examples are given to illustrate this comparison our results can be viewed as important refinement and improvement of the previously known results

a geometric construction of iterative functions of order three to solve nonlinear equations in this paper we consider a geometric construction of iteration functions of order three to develop cubically convergent iterative methods for solving nonlinear equations this construction can be applied to any iteration function of order two to develop an iteration function of order three some examples are given of deriving several third order iteration methods and several numerical results follow to illustrate the performance of the derived methods