a general central limit theorem for fkg systems a central limit theorem is given which is applicable to not necessarily monotonic functions of random variables satisfying the fkg inequalities one consequence is convergence of the block spin scaling limit for the magnetization and energy densities jointly to the infinite temperature fixed point of independent gaussian blocks for a large class of ising ferromagnets whenever the susceptibility is finite another consequence is a central limit theorem for the density of thesurface of the infinite cluster in percolation models

the theory of large deviations from boltzmann s 1877 calculation to equilibrium macrostates in 2d turbulence abstract after presenting some basic ideas in the theory of large deviations this paper applies the theory to a number of problems in statistical mechanics these include deriving the form of the gibbs state for a discrete ideal gas describing probabilistically the phase transition in the curie weiss model of a ferromagnet and deriving variational formulas that describe the equilibrium macrostates in models of two dimensional turbulence a general approach to the large deviation analysis of models in statistical mechanics is also formulated

a family of second derivative free fourth order continuation method for solving nonlinear equations in this paper we present a parameter based iterative method free from the second derivative for solving nonlinear equations of type f x 0 ezquerro and hernandez 1999 discussed the convergence analysis of a uniparametric family of an iterative method in banach space based on this idea we propose a uniparametric family of second derivative free iterative method in r we discuss the convergence analysis and observe that it possesses a fourth order convergence for solving nonlinear equations in r several numerical examples are worked out with our proposed methods for parameter 1 2 and the existing fourth order iterative method proposed in king 1973 chun 2007 soleymani et al 2012 and maheshwari 2009 finally from the comparison we observe that our method is more efficient than existing methods finally we compare basins of attraction of our methods with the second derivative free fourth order iterative method proposed in king 1973 chun 2007 soleymani et al 2012 and maheshwari 2009 observe that the proposed scheme is more efficient

analysis of two chebyshev like third order methods free from second derivatives for solving systems of nonlinear equations in this paper two chebyshev like third order methods free from second derivatives are considered and analyzed for systems of nonlinear equations the methods can be obtained by having different approximations to the second derivatives present in the chebyshev method we study the local and third order convergence of the methods using the point of attraction theory the computational aspects of the methods are also studied using some numerical experiments including an application to the chandrasekhar integral equations in radiative transfer

quasi sure analysis and stratonovich anticipative stochastic differential equations consider a stochastic differential equation on d with smooth and bounded coefficients we apply the techniques of the quasi sure analysis to show that this equation can be solved pathwise out of a slim set furthermore we can restrict the equation to the level sets of a nondegenerate and smooth random variable and this provides a method to construct the solution to an anticipating stochastic differential equation with smooth and nondegenerate initial condition

complete equivalence of the gibbs ensembles for one dimensional markov systems we show the equivalence of the gibbs ensembles at the level of measures for one dimensional markov systems with arbitrary boundary conditions that is the limit of the microcanonical gibbs ensemble is a gibbs measure with an interaction depending on the microcanonical constraint in fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins including the boundary spins are fixed when conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble we show that the limit is a gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures for this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions we establish analogous results for finite range potentials

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

fourth order iterative methods free from second derivative in this paper we present a family of new iterative methods with order of convergence four for solving non linear equations per iteration these methods require one evaluation of the function and two of its first derivative analysis of efficiency in term of function evaluations shows that this family of methods has great superiority which is also demonstrated by numerical examples

galton watson trees with vanishing martingale limit we show that an infinite galton watson tree conditioned on its martingale limit being smaller than varepsilon agrees up to generation k with a regular mu ary tree where mu is the essential minimum of the offspring distribution and the random variable k is strongly concentrated near an explicit deterministic function growing like a multiple of log 1 varepsilon more precisely we show that if mu ge 2 then with high probability as varepsilon downarrow 0 k takes exactly one or two values this shows in particular that the conditioned trees converge to the regular mu ary tree providing an example of entropic repulsion where the limit has vanishing entropy our proofs are based on recent results on the left tail behaviour of the martingale limit obtained by fleischmann and wachtel 11

malliavin calculus for infinite dimensional systems with additive noise abstract we consider an infinite dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of wiener processes by studying how randomness is spread by the dynamics we develop in this setting a partial counterpart of hormander s classical theory of hypoelliptic operators we study the distributions of finite dimensional projections of the solutions and give conditions that provide existence and smoothness of densities of these distributions with respect to the lebesgue measure we also apply our results to concrete spdes such as a stochastic reaction diffusion equation and the stochastic 2d navier stokes system

pseudocomposition a technique to design predictor corrector methods for systems of nonlinear equations abstract a new technique for designing iterative methods for solving nonlinear systems is presented this procedure called pseudocomposition uses a known method as a predictor and the gaussian quadrature as a corrector the order of convergence of the resulting scheme depends among other factors on the order of the last two steps of the predictor we also introduce a new iterative algorithm of order six and apply the mentioned technique to generate a new method of order ten finally some numerical test are shown

improved chebyshev halley methods with sixth and eighth order convergence we present a second derivative free family of modified chebyshev halley methods with increasing order of convergence for solving nonlinear equations the idea is based on the recent development by li et al 2014 analysis of convergence shows that the family possesses at least sixth order of convergence in a particular case even eighth order of convergence is achieved per iteration each method of the family requires four evaluations namely three functions and one first derivative that means the eighth order method is optimal in the sense of kung traub hypothesis numerical tests are performed which confirm the theoretical results regarding order of convergence and efficiency of the new methods

a uniparametric chebyshev type method free from second derivatives in this paper we present a family of new chebyshev type methods free from second derivatives for solving non linear equations analysis of convergence shows that these new methods are cubically convergent as particular cases we introduce two efficient methods in this family of the new methods the practical utility of which is demonstrated by numerical examples

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

entropic repulsion of the lattice free field consider the massless free field on thed dimensional lattice d d 3 that is the centered gaussian field on open image in new window with covariances given by the green function of the simple random walk on d we show that the probability that all the spins are positive in a box of volumen d decays exponentially at a rate of ordernd 2 logn and compute explicitly the corresponding constant in terms of the capacity of the unit cube the result is extended to a class of transient random walks with transition functions in the domain of the normal and stable law

large deviations for ℤ d actions we establish large deviations bounds for translation invariant gibbs measures of multidimensional subshifts of finite type this generalizes fo and partially c o and b where only full shifts were considered our framework includes in particular the hard core lattice gas models which are outside of the scope of fo c o and b

modified halley s method free from second derivative in this paper we derive a class of new third order methods free from second derivative from halley s method per iteration the methods require two evaluations of the function and one evaluation of its first derivative analysis of efficiency in term of function evaluations of this class of methods shows that they have definite practical utility which is also demonstrated by numerical examples

an improvement of chebyshev halley methods free from second derivative abstract in this paper a family of modified chebyshev halley s methods free from second derivative is presented per iteration the new methods require three evaluations of the function and one of its first derivatives a detailed convergence analysis of the new methods shows that the new methods are at least fifth order convergent and especially the modified super halley s method is sixth order convergent numerical examples are given to illustrate the efficiency and performance of the new methods

decoupling of banach valued multilinear forms in independent symmetric banach valued random variables let e be a banach space and e be symmetric continuous and convex let ui and ri be independent sequences of random variables having respectively u 0 1 and symmetric bernoulli distributions and let u i j and r i j for j 1 2 d be independent copies of these sequences we prove two sided inequalities between the quantities e phi sum limits underset raise0 3em hbox smash scriptscriptstyle i in mathbb z d r i 1 r i d f underset raise0 3em hbox smash scriptscriptstyle i u i 1 u i d and their decoupled versions e phi sum limits underset raise0 3em hbox smash scriptscriptstyle iota in mathbb z d r i 1 1 r i d d f underset raise0 3em hbox smash scriptscriptstyle iota u i 1 1 u i d d for bochner integrable f i 0 1 d e this generalizes results of kwapien and of zinn

semilocal convergence of a family of third order methods in banach spaces under holder continuous second derivative abstract in this paper the semilocal convergence of a family of multipoint third order methods used for solving f x 0 in banach spaces is established it is done by using recurrence relations under the assumption that the second frechet derivative of f satisfies holder continuity condition based on two parameters depending upon f a new family of recurrence relations is defined using these recurrence relations an existence uniqueness theorem is established to prove that the r order convergence of the method is 2 p a priori error bounds for the method are also derived two numerical examples are worked out to demonstrate the efficacy of our approach

malliavin calculus and decoupling inequalities in banach spaces abstract we develop a theory of malliavin calculus for banach space valued random variables using radonifying operators instead of symmetric tensor products we extend the wiener ito isometry to banach spaces in the white noise case we obtain two sided l p estimates for multiple stochastic integrals in arbitrary banach spaces it is shown that the malliavin derivative is bounded on vector valued wiener ito chaoses our main tools are decoupling inequalities for vector valued random variables in the opposite direction we use meyer s inequalities to give a new proof of a decoupling result for gaussian chaoses in umd banach spaces

zero finder methods derived from obreshkov s techniques in this paper two families of zero finding iterative methods for solving nonlinear equations f x 0 are presented the key idea to derive them is to solve an initial value problem applying obreshkov like techniques more explicitly obreshkov s methods have been used to numerically solve an initial value problem that involves the inverse of the function f that defines the equation carrying out this procedure several methods with different orders of local convergence have been obtained an analysis of the efficiency of these methods is given finally we introduce the concept of extrapolated computational order of convergence with the aim of numerically test the given methods a procedure for the implementation of an iterative method with an adaptive multi precision arithmetic is also presented

a note on the halley method in banach spaces abstract we provide a convergence theorem under the usual ostrowski kantorovich conditions for an equivalent form of the halley method in banach spaces the best possible upper error bound for the method is obtained some applications to the solution of quadratic operator equations are provided

a composite third order newton steffensen method for solving nonlinear equations in this paper we suggest a third order method formed by the composition of newton and steffensen methods for finding simple and real roots of a nonlinear equation in single variable per iteration the formula requires two evaluations of the function and single evaluation of the derivative experiments show that the method is suitable in the cases where newton and steffensen methods fail

a new modified halley method without second derivatives for nonlinear equation in a recent paper noor and noor k inayat noor m aslam noor predictor corrector halley method for nonlinear equations appl math comput in press doi 10 1016 j amc 11 023 have suggested and analyzed a predictor corrector method halley method for solving nonlinear equations in this paper we modified this method by using the finite difference scheme which has a quintic convergence we have compared this modified halley method with some other iterative of fifth orders convergence methods which shows that this new method is a robust one several examples are given to illustrate the efficiency and the performance of this new method

a simple and efficient method with high order convergence for solving systems of nonlinear equations we propose an m 1 step modified newton method of convergence order m 2 to solve systems of nonlinear equations which are third frechet differentiable in a convex set containing the zero computational efficiency in the general form for a positive integer m is discussed which shows that the efficiency increases with m when applied to large systems moreover a comparison between the efficiency of this technique and some existing efficient methods is made which implies that the present method is more efficient particularly for solving large systems of equations theoretical results about order of convergence and computational efficiency are largely verified in numerical examples

an optimal reconstruction of chebyshev halley type methods for nonlinear equations having multiple zeros abstract establishment of a new optimal higher order iterative scheme for multiple zeros with known multiplicity m 1 of univariate function is one of the hard and demanding task in the area of computational mathematics and numerical analysis we write this paper with the aim to propose a new higher order reconstruction of chebyshev halley type iteration functions for multiple zeros in a simple way minimum order of convergence of our scheme is seven and maximum reaches to eight without using any extra functional evaluation with regard to computational cost each member of our scheme needs four functional evaluations at each step so the maximum efficiency index of our scheme is 1 6818 for 1 in addition we thoroughly study the convergence analysis which confirms the theoretical order of convergence moreover we check the efficiency and compare them to the existing ones on four real life problems namely fractional conversion chemical engineering continuous stirred tank reactor and eigen value problems and two well known academic problems finally we confirm on the basis of obtained results that our methods have better residual error errors between two consecutive iterations and stable computational order of convergence as compared to the other existing ones

self similarity of brownian motion and a large deviation principle for random fields on a binary tree using self similarity of brownian motion and its representation as a product measure on a binary tree we construct a random sequence of probability measures which converges to the distribution of the brownian bridge we establish a large deviation principle for random fields on a binary tree this leads to a class of probability measures with a certain self similarity property the same construction can be carried out forc 0 1 valued processes and we can describe for instance ac 0 1 valued ornstein uhlenbeck process as a large deviation of brownian sheet

some second derivative free variants of halley s method for multiple roots in this paper we present two new families of third order methods for finding multiple roots of nonlinear equations each of them is based on a variant of the halley s method for simple roots free from second derivative one of the families requires one evaluation of the function and two of its first derivative per iteration and the other family requires two evaluations of the function and one of its first derivative several numerical examples are given to illustrate the performance of the presented methods

a family of optimal three point methods for solving nonlinear equations using two parametric functions abstract using an interactive approach which combines symbolic computation and taylor s series a wide family of three point iterative methods for solving nonlinear equations is constructed these methods use two suitable parametric functions at the second and third step and reach the eighth order of convergence consuming only four function evaluations per iteration this means that the proposed family supports the kung traub hypothesis 1974 on the upper bound 2 m of the order of multipoint methods based on m 1 function evaluations providing very high computational efficiency different methods are obtained by taking specific parametric functions the presented numerical examples demonstrate exceptional convergence speed with only few function evaluations

steffensen type methods for solving nonlinear equations in the present paper by approximating the derivatives in the well known fourth order ostrowski s method and in a sixth order improved ostrowski s method by central difference quotients we obtain new modifications of these methods free from derivatives we prove the important fact that the methods obtained preserve their convergence orders 4 and 6 respectively without calculating any derivatives finally numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical newton s method

microcanonical distributions for lattice gases in this article a large deviation principle cf theorem 1 3 for the empirical distribution functional is applied to prove a rather general version of boltzmann s principle cf theorem 3 5 for models with shift invariant finite range potentials the final section contains an application of these considerations to the two dimensional ising model at sub critical temperature

decoupling inequalities for multilinear forms in independent symmetric random variables on considere x 1 x 2 des copies independantes d une suite x x 1 x 2 de variables aleatoires symetriques independantes et m une forme multilineaire symetrique de rang s sur r n dont les composantes a i1 a is relatives a la base standard de r n satisfont a i1 i s 0

on the convergence of a higher order family of methods and its dynamics in this paper we present the study of the local convergence of a higher 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 by means of studying the associated rational function parameter spaces are shown and the study of the stability of all the fixed points is presented

an acceleration of newton s method super halley method from a study of the convexity we give an acceleration for newton s method and obtain a new third order method then we use this method for solving non linear equations in banach spaces establishing conditions on convergence existence and uniqueness of solution as well as error estimates

an improvement of the euler chebyshev iterative method abstract we present a new method for the computation of the solutions of nonlinear equations when it is necessary to use high precision we improve the euler chebyshev iterative method which is a generalization of an improvement of newton s method a symbolic computation allows us to find the best coefficients respect to the local order of convergence the adaptation of the strategy presented here gives an additional iteration function with an additional evaluation of the function it provides a lower cost if we use adaptive multi precision arithmetics the numerical results computed using this system with a floating point representing a maximum of 210 decimal digits support this theory

on a family of second derivative free variants of chebyshev s method in this paper we present a family of new second derivative free variants of chebyshev s method in which f x 0 in some points is permitted analysis of convergence shows that the new methods are cubically convergent analysis of efficiency in term of function evaluations shows that the new methods have the great practical utility which is demonstrated by numerical examples

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

some second derivative free variants of chebyshev halley methods in this paper we present some new variants of the chebyshev halley methods free from second derivative the new methods require two function and one first derivative evaluations per iteration and are shown to be third order convergent several numerical examples are given to illustrate the performance of the presented methods

the construction of rational iterative methods for solving nonlinear equations abstract in this paper based on the idea of constructing iterative methods by sharma j r sharma a family of third order methods to solve nonlinear equations appl math comput 184 2007 210 215 jiang and han dongdong jiang danfu han some one parameter families of third order methods for solving nonlinear equations appl math comput 195 2008 392 396 we propose some rational iterative methods by linearizing the quadratic curve equation by analysis of asymptotic errors we prove the new families are all cubically convergent as an example we present a new concrete iterative family with better convergent performance than classic third order methods through the adjustment of parameter n automatically

dynamics of a family of chebyshev halley type methods in this paper the dynamics of the chebyshev halley family is studied on quadratic polynomials a singular set that we call cat set appears in the parameter space associated to the family this set has interesting similarities with the mandelbrot set the parameter space has allowed us to find different elements of the family which have bad convergence properties since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method

construction of optimal order nonlinear solvers using inverse interpolation abstract there is a vast literature on finding simple roots of nonlinear equations by iterative methods these methods can be classified by order by the information used or by efficiency there are very few optimal methods that is methods of order 2 m requiring m 1 function evaluations per iteration here we give a general way to construct such methods by using inverse interpolation and any optimal two point method the presented optimal multipoint methods are tested on numerical examples and compared to existing methods of the same order of convergence

entropy for translation invariant random cluster measures we study translation invariant random cluster measures with techniques from large deviation theory and convex analysis in particular we prove a large deviation principle with rate function given by a specific entropy and a dobrushinlanfordruelle variational principle that characterizes translation invariant random cluster measures as the solutions of the variational equation for free energy consequences of these theorems include inequalities for edge and cluster densities of translationinvariant random cluster measures 1 introduction the random cluster model was introduced around 1972 by fortuin and kasteleyn 18 as a family of models that contains the bernoulli bond percolation model the ising model and the q state potts models as special cases it has developed into an important and very successful tool for studying the ising and potts models our paper focuses on the properties of the random cluster measures themselves our starting point is grimmett s 23 development of elements of a thermodynamic formalism for this model beginning with a definition of infinite volume random cluster measures in terms of a dlr dobrushinlanfordruelle equation just as for gibbsian systems natural next questions concern the validity of large deviation principles and variational characterizations of infinite volume measures these questions we answer in the present paper and derive some consequences here is a brief overview of the dlr approach of 23 the random cluster measures of interest live on the graph formed by putting edges between nearest neighbor vertices of the d dimensional hypercubic lattice d the edge that connects two vertices x and y that are l 1 distance one apart is

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

sobolev spaces of banach valued functions associated with a markov process we discuss sobolev spaces of banach valued functions they are extensions of sobolev spaces of scalar functions we use a gamma transform of a semigroup associated with a markov process a typical example is the ornstein uhlenbeck process on the wiener space

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

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

modification of the kantorovich assumptions for semilocal convergence of the chebyshev method this study obtains two semilocal convergence results for the well known chebyshev method which is a third order iterative process the hypotheses required are modifications to the normal kantorovich ones the results obtained are applied to the reduction of nonlinear integral equations of the fredholm type and first kind

large deviations and the maximum entropy principle for marked point random fields we establish large deviation principles for the stationary and the individual empirical fields of poisson and certain interacting random fields of marked point particles in d the underlying topologies are induced by a class of not necessarily bounded local functions and thus finer than the usual weak topologies our methods yield further that the limiting behaviour of conditional poisson distributions as well as certain distributions of gibbsian type is governed by the maximum entropy principle we also discuss various applications and examples

entropy limit theorems and variational principles for disordered lattice systems we study infinite volume limits and gibbs states of disordered lattice systems with bounded and continuous potentials our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes we find that many familiar results of invariant potentials such as large deviation theorems variational principles and equivalence of ensembles continue to hold for disordered models with suitably modified statements

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

the jarratt method in banach space setting abstract in this note we extend the jarratt method of order four into banach spaces we also establish a kantorovich type convergence theorem see the references and give an explicit expression for the error bound of the method jarratt 1966 1969

stochastic integration in umd banach spaces in this paper we construct a theory of stochastic integration of processes with values in l h e where h is a separable hilbert space and e is a umd banach space i e a space in which martingale differences are unconditional the integrator is an h cylindrical brownian motion our approach is based on a two sided l p decoupling inequality for umd spaces due to garling which is combined with the theory of stochastic integration of l h e valued functions introduced recently by two of the authors we obtain various characterizations of the stochastic integral and prove versions of the ito isometry the burkholder davis gundy inequalities and the representation theorem for brownian martingales

on the local convergence and the dynamics of chebyshev halley methods with six and eight order of convergence we study the local convergence of chebyshev halley methods with six and eight order of convergence to approximate a locally unique solution of a nonlinear equation in sharma 2015 see theorem 1 p 121 the convergence of the method was shown under hypotheses reaching up to the third derivative the convergence in this study is shown under hypotheses on the first derivative hence the applicability of the method is expanded the dynamics of these methods are also studied finally numerical examples examining dynamical planes are also provided in this study to solve equations in cases where earlier studies cannot apply

higher order derivative free families of chebyshev halley type methods with or without memory for solving nonlinear equations abstract in this paper we present two new derivative free families of chebyshev halley type methods for solving nonlinear equations numerically both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence furthermore accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step the self accelerating parameter is estimated from the current and previous iteration this self accelerating parameter is calculated using newton s interpolation polynomial of third and fourth degrees consequently the r orders of convergence are increased from 4 to 6 and 8 to 12 respectively without any additional functional evaluation the results require high order derivatives reaching up to the eighth derivative that is why we also present an alternative approach using only the first or at most the fourth derivative we also obtain the radius of convergence and computable error bounds on the distances involved numerical experiments and the comparison of the existing robust methods are included to confirm the theoretical results and high computational efficiency in particular we consider a concrete variety of real life problems coming from different disciplines e g kepler s equation of motion planck s radiation law problem fractional conversion in a chemical reactor the trajectory of an electron in the air gap between two parallel plates van der waal s equation which explains the behavior of a real gas by introducing in the ideal gas equations in order to check the applicability and effectiveness of our proposed methods

a family of modified super halley methods with fourth order convergence in this paper we present a family of modified super halley methods for solving non linear equations analysis of convergence shows that the methods have fourth order convergence the superiority of the new methods is that they require no additional evaluations of the function the first derivative or second derivative as compared with the classical third order methods although their order is improved numerical results show that the new methods can be efficient

a new modified secant like method for solving nonlinear equations in this paper we present a new secant like method for solving nonlinear equations analysis of the convergence shows that the asymptotic convergence order of this method is 1 3 some numerical results are given to demonstrate its efficiency

on tests for normality the problem of deciding whether a sample of a random field was generated by a gaussian distribution is considered based on extensions of large deviation estimates due to m d donsker and s r s varadhan 1985 a test that is optimal in a generalized neyman pearson sense is proposed this test turns out to depend on properties of the entropy of gaussian processes and does not depend on cumulant computations

some variants of chebyshev halley methods free from second derivative abstract in this paper we present some new modifications of chebyshev halley methods free from second derivative the convergence analysis shows that the new methods are third order convergent and require one function and two first derivative evaluations per iteration several numerical examples are given to illustrate the performance of the presented methods

a method for finding sharp error bounds for newton s method under the kantorovic assumptions this paper gives a method for finding sharpa posteriori error bounds for newton s method under the assumptions of kantorovich s theorem on the basis of this method new error bounds are derived and comparison is made among the known bounds of dennis 2 doring 4 gragg tapia 5 kantorovich 6 7 kornstaedt 9 lancaster 10 miel 11 13 moret 14 ostrowski 17 18 potra 19 and potra ptak 20

decoupling inequalities for the tail probabilities of multivariate u statistics in this paper we present a decoupling inequality that shows that multivariate u statistics can be studied as sums of conditionally independent random variables this result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration more precisely we get the following result let x j be a sequence of independent random variables on a measurable space mathscr j s and let x j i j 1 ldots k be k independent copies of x i let f i 1i 2 ldots i k be families of functions of k variables taking s times cdots times s into a banach space b cdots then for all n geq k geq 2 t 0 there exist numerical constants c k depending on k only so that p bigg big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 1 i 2 ldots x 1 i k big geq t bigg leq c kp bigg c k big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 2 i 2 ldots x k i k big geq t bigg the reverse bound holds if in addition the following symmetry condition holds almost surely f i 1i 2 cdots i k x i 1 x i 2 ldots x i k f i pi 1 i pi 2 cdots i pi k x i pi 1 x i pi 2 ldots x i pi k for all permutations pi of 1 ldots k

a family of three point methods of optimal order for solving nonlinear equations a family of three point iterative methods for solving nonlinear equations is constructed using a suitable parametric function and two arbitrary real parameters it is proved that these methods have the convergence order eight requiring only four function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 3 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class of root solvers possesses very high computational efficiency numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

a non iterative method for solving non linear equations abstract in this paper using a hyperbolic tangent function tanh x 0 we develop a non iterative method to estimate a root of an equation f x 0 the problem of finding root is transformed to evaluating an integral and thus we need not take account of choosing initial guess the larger the value of the better the approximation to the root alternatively we employ the signum function sgn x instead of the hyperbolic tangent function which results in an exact formula for the root availability of the present method is shown by some numerical examples for which the traditional newton s method is not appropriate

stochastic evolution equations in umd banach spaces we discuss existence uniqueness and space time holder regular ity for solutions of the parabolic stochastic evolution equation du t au t f t u t dt b t u t dwh t t 2 0 t0 u 0 u0 where a generates an analytic c0 semigroup on a umd banach space e and wh is a cylindrical brownian motion with values in a hilbert space h we prove that if the mappings f 0 t e e and b 0 t e l h e satisfy suitable lipschitz conditions and u0 is f0 measurable and bounded then this problem has a unique mild solution which has trajectories in c 0 t d a provided 0 and 0 satisfy 1 2 vari ous extensions of this result are given and the results are applied to parabolic stochastic partial dierential equations

decoupling inequalities for polynomial chaos let x x 1 ldots x d be a sequence of independent symmetric identically distributed random vectors with independent components the main subject of this paper is the so called decoupling inequalities i e inequalities of the form begin align e phi cq x x ldots x leq e phi q x 1 x 2 ldots x d leq e phi cq x x ldots x end align where q is a symmetric multilinear form with values in a vector space f with all diagonal terms equal to zero and phi is a convex function on f

some second derivative free variants of super halley method with fourth order convergence in this paper we present some new variants of super halley method for solving nonlinear equations these methods are free from second derivatives and require one function and two first derivative evaluations per iteration analysis of convergence shows that the methods are fourth order several numerical examples are given to illustrate the performance of the presented methods

on a general class of multipoint root finding methods of high computational efficiency a general class of n point iterative methods for solving nonlinear equations is constructed by combining methods of newton s type and an arbitrary two point method of the fourth order of convergence it is proved that these methods have the convergence order 2 n requiring only n 1 function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class possess as high as possible computational efficiency in the sense of the kung traub hypothesis numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

modifications of higher order convergence for solving nonlinear equations abstract in liang fang guoping he some modifications of newton s method with higher order convergence for solving nonlinear equations j comput appl math 228 2009 296 303 the authors pointed out that the iteration constructed in y m ham c b chun and s g lee some higher order modifications of newton s method for solving nonlinear equations j comput appl math 222 2008 477 486 failed when p 2 they gave some counterexamples and obtained a modified result however they did not show the essential reason which leads to the incorrect result in this paper we shall show that reason and present more general results than the above mentioned papers

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

ball convergence comparison between three iterative methods in banach space under hypothese only on the first derivative we present a convergence ball comparison between three iterative methods for approximating a locally unique solution of a nonlinear equation in a banach space setting the convergence ball and error estimates are given for these methods under hypotheses only on the first frechet derivative in contrast to earlier studies such as adomian 1994 1 babajee et al 2008 13 cordero and torregrosa 2007 17 cordero et al 18 darvishi and barati 2007 19 using hypotheses reaching up to the fourth frechet derivative although only the first derivative appears in these methods this way we expand the applicability of these methods numerical examples are also presented in this study

large deviations and maximum entropy principle for interacting random fields on mathbb z d we present a new approach to the principle of large deviations for the empirical field of a gibbsian random field on the integer lattice zd this approach has two main features first we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities and thus obtain estimates which are finer and more widely applicable second we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field the limits being those predicted by the maximum entropy principle this result implies a general version of the equivalence of gibbs ensembles stating that every microcanonical limiting state is a grand canonical equilibrium state we also prove a converse to the last statement and discuss some applications 0 introduction as is well known the study of the asymptotic probabilities of large fluctuations of time averages or space averages away from the mean is based on two fundamental principles the principle of large deviations and the maximum entropy principle the former provides the exact rate of exponential decay of the fluctuation probabilities whereas the latter predicts the limiting conditional distribution under the condition that the fluctuations are large it is obvious that these principles are intimately related in this paper we investigate these principles in the case of interacting random fields on the integer lattice 7zd the setup is the following first we let e 4 be any measurable space we shall assume throughout that e a is standard borel but we shall avoid making any explicit topological assumptions on e so we do not assume that e is polish next we let s zd be the d dimensional integer lattice and q f e gas the associated product space in lattice models of statistical mechanics e is called the state space or single spin space and q f the configuration space we let j s f denote the set of all probability

modified chebyshev halley methods free from second derivative abstract in this paper we present some variants of chebyshev halley methods free from second derivative it is shown that the order of convergence of the proposed methods is three at least there are two functions and one first derivative evaluations per iteration several numerical examples are given to demonstrate the performance of the presented methods by comparing with some other methods

modified chebyshev s method free from second derivative for non linear equations in this paper we present some new modifications of chebyshev s method free from second derivative the order of convergence of the new methods is three per iteration the methods require two evaluations of the function and one evaluation of its first derivative the convergence of the new methods for the case of multiple roots is also discussed numerical examples show that the new methods have definite practical utility

large deviation principles for euclidean functionals and other nearly additive processes we prove a large deviation principle for a process indexed by cubes of the multidimensional integer lattice or euclidean space under approximate additivity and regularity hypotheses the rate function is the convex dual of the limiting logarithmic moment generating function in some applications the rate function can be expressed in terms of relative entropy the general result applies to processes in euclidean combinatorial optimization statistical mechanics and computational geometry examples include the length of the minimal tour the traveling salesman problem the length of the minimal matching graph the length of the minimal spanning tree the length of the k nearest neighbors graph and the free energy of a short range spin glass model

the divergence of banach space valued random variables on wiener space the domain of definition of the divergence operator on an abstract wiener space w h is extended to include w valued and open image in new window valued integrands the main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in s extended domain have divergence zero these results are then applied to the analysis of quasiinvariant flows induced by w valued vector fields and among other results it turns out that these divergence free vector fields are responsible for generating measure preserving flows

on halley type iterations with free second derivative in this paper we relax the convergence conditions required in ezquerro and hernandez int j pure appl math 6 1 2003 103 for a multipoint third order iteration of halley type where the conditions provided are the known ones for methods of order three

modified efficient variant of super halley method in this paper we present a new modified variant of super halley method for solving non linear equations numerical results show that the method has definite practical utility