iteration functions re visited two classes of iteration functions ifs are derived in this paper the first one point ifs was originally derived by joseph traub using a different approach to ours simultaneous ifs the second is new and is demonstrably shown to be more informationally efficient than the first these ifs apply to polynomials with arbitrary complex coefficients and zeros which can also be multiple

a high order iterative formula for simultaneous determination of zeros of a polynomial abstract we propose a hybrid method to determine all the zeros of a polynomial simultaneously by combining the single root method and the simultaneous one the present method has a high convergence order even for multiple roots by using the pade approximation

a note on determinantal representation of a schroder konig like simultaneous method for finding polynomial zeros abstract using pade approximation sakurai torii and sugiura derived in the paper sakurai et al 1991 the generalized iterative method of order n 2 for finding all zeros of a polynomial where n is the highest order of a polynomial derivative involved in the presented iterative formula in this note we give the determinantal representation of this method and analyze procedures for its implementation and some computational aspects

accelerated convergence in newton s method newton s method is based on a linear approximation of the function whose roots are to be determined taken at the current point and the resulting algorithm is known to converge quadratically in a procedure to increase the rate of convergence the author modifies the target function in such a way that newton s method applied to the modified function will yield a faster rate of convergence

on rediscovered methods for solving equations abstract in the recent papers of gerlach siam rev 36 1994 272 276 and ford and pennline siam rev 38 1996 658 659 a class of iterative methods for solving a single equation f x 0 with arbitrary rate of convergence has been presented in this paper we show that this class is equivalent to five other classes of iterative methods derived earlier in various ways and expressed in different forms the proofs of equivalence of all considered iteration formulas are given

global properties of supermanifolds we construct new examples of supermanifolds and determine the vector bundle structure of the supermanifolds commonly used in physics we show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts and which is one of a nested sequence of foliations we point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds for example a compact supermanifold with a single even dimension must have vanishing euler characteristic we also show that a globally defined superfield on a nice compact supermanifold must be constant along the leaves of the foliations by this mechanism the global topology of a supermanifold can be used to impose physically interesting constraints on superfields as an example we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral

the theory of g supermanifolds on developpe une theorie des supervarietes dans laquelle une supervariete est une variete ordinaire associee a une g structure d ordre 2 integrable on demontre un theoreme de structure

on some simultaneous methods based on weierstrass correction abstract the purpose of the paper is to present some procedures for the construction of simultaneous methods for finding all zeros of a polynomial f using weierstrass corrections w i z f z k i z z k in this manner some well known methods as well as new methods are obtained the proposed iterative algorithms are realized in ordinary complex arithmetic and circular interval arithmetic the convergence analysis of the presented methods and numerical examples are given

a general help inequality connected with symmetric operators in this paper a general help hardy everitt littlewood polya inequality is considered which is connected with a symmetric operator in a hilbert space and abstract boundary mappings a criterion for the validity of such an inequality in terms of the abstract titchmarsh weyl function is proved and applied to sturm liouville operators difference operators a hamiltonian system and a block operator matrix

orthogonal polynomials and extensions of copson s inequality this is a survey of recent results on a class of series inequalities involving second order difference operators which includes a well known inequality of copson s a connection has been established between these inequalities and the properties of the hellinger nevanlinna m function for an associated recurrence relation mx aw x a e c the validity of the inequality the value of the best constant and the nontrivial equalising sequences if they exist are determined in terms of m the function m is shown to have an integral representation in terms of a measure with respect to which polynomial solutions of the recurrence relation are orthogonal and this is used to examine a number of examples of the inequality the best constants in some cases have only been evaluated numerically

on a series inequality for limit circle and finite problems brown and evans proposed a criterion for the validity of their series analogue of everitt s extension of the hardy littlewood inequality presupposes that the associated difference equation is in the strong limit point at infinity in this paper we investigate the case when the difference equation involved is in the limit circle case and non oscillatory and also when the interval is finite

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

a return to the hardy littlewood integral inequality this paper is concerned with the extension and generalization of a classic integral inequality first considered by hardy and littlewood in 1932 the integral inequality is big int b a p f 2 q f 2 big 2 leqslant k int b a w f 2 int b a w w 1 pf qf 2 f in delta where k is a positive number the coefficients p q and w are real valued and w 0 on the interval a b infty a b leqslant infty and p 1 q w in l loc a b delta is the maximal linear manifold of functions f for which the two integrals on the right hand side are both finite the hardy littlewood inequality is the special case a 0 b infty p w 1 and q 0 on 0 infty here k 4 and all the cases of equality can be identified in this paper the results of everitt in 1971 connecting the above inequality with the titchmarsh weyl m coefficient of the associated differential equation py qy lambda wy on a b are restated in an improved form and are shown to hold under the above minimal conditions on the coefficients p q and w it is proved that the best possible value for the number k satisfies 1 k leqslant infty and that within these bounds all values of k can be realized in the course of establishing these results new information of independent interest is obtained about the behaviour of the m coefficient near the real axis examples are given to exhibit the wide range of possibilities for the integral inequality recent results on the asymptotic form of the m coefficient and on the transformation theory of the associated differential equation are shown to have applications for the general properties of the inequality all recent contributions to this general integral inequality are listed in the references and their content is surveyed during the course of the paper

newton s method and generation of a determinantal family of iteration functions abstract it is well known that halley s method can be obtained by applying newton s method to the function f f gerlach siam rev 36 1994 272 276 gives a generalization of this approach and for each m 2 recursively defines an iteration function gm x having order m kalantari et al j comput appl math 80 1997 209 226 and kalantari technical report dcs tr 328 department of computer science rutgers university new brunswick nj 1997 derive and characterize a determinantal family of iteration functions called the basic family bm x m 2 in this paper we prove gm x bm x on the one hand this implies that gm x enjoys the previously derived properties of bm x i e the closed formula its efficient computation an expansion formula which gives its precise asymptotic constant as well as its multipoint versions on the other this gives a new insight on the basic family and newton s method

construction of zero finding methods by weierstrass functions abstract in this paper we present a simple and elegant procedure for the construction of iterative methods for the simultaneous determination of simple or multiple zeros of an algebraic polynomial this procedure is based on the application of a special type of functions called weierstrass functions to suitable zero finding methods for a single zero for demonstration using this approach we derive many known iterative methods in a simpler way compared with original derivations as well as some new methods aside from the presented methodology in developing zero finding methods the paper offers a short review of simultaneous methods including some historical notes

how different are the supermanifolds of rogers and dewitt a dewitt supermanifold always has the structure of a vector bundle over an ordinary spacetime manifold whereas a rogers supermanifold is not so restricted corresponding to the vector space fibers of the dewitt supermanifold a rogers supermanifold has a foliation by submanifolds or leaves parametrized by soul coordinates only we show that the universal covering space of any leaf always admits a flat metric if the covering space is complete in this metric it must in fact be a vector space we combine this result with known theorems about foliations to give conditions under which a compact rogers supermanifold with a single even dimension is necessarily a quotient space of flat superspace we also show that a supermanifold defined by a polynomial equation in flat superspace is always of the dewitt type finally we exhibit new supermanifold structures forr2 and the 2 torus which show that the foliation of a rogers supermanifold can be quite exotic

discrete inequalities orthogonal polynomials and the spectral theory of difference operators in a recent paper the first two authors studied a class of series inequalities associated with a three term recurrence relation which includes a well known inequality of copson s it was shown that the validity of the inequality and the value of the best constant are determined in terms of the so called hellinger nevanlinna m function the theory is the discrete analogue of that established by everitt for a class of integro differential inequalities in this paper the properties of the m function are investigated and connections with the theory of orthogonal polynomials and the hamburger moment problem are explored the results are applied to give examples of the series inequalities associated with the classical orthogonal polynomials

a new fourth order family of simultaneous methods for finding polynomial zeros using a suitable fixed point relation with a complex parameter a new one parameter family of simultaneous methods of the fourth order for finding complex zeros of a polynomial is derived in ordinary complex arithmetic convergence analysis of the presented family is performed under computationally verifiable initial conditions which depend only on polynomial coefficients and initial approximations further improvements of the proposed family of methods are discussed and a modification for finding multiple zeros is presented some numerical results for various values of the parameter are given

semilocal convergence of chebyshev like root finding method for simultaneous approximation of polynomial zeros abstract in this paper we present convergence results for the chebyshev like method for the simultaneous computation of all zeros of a polynomial f over a complete normed field our results generalize improve and complement the result of petkovic and petkovic 2001 10 the new results give weaker sufficient convergence conditions a priori and a posteriori error estimates as well as information on the location of the zeros another important aspect of this work is that we do not assume neither simplicity nor existence of the zeros of f furthermore several numerical examples are provided to show some practical applications of our results

on an extension of copson s inequality for infinite series in 1979 copson proved the following analogue of the hardy littlewood inequality if is a sequence of real numbers such that are convergent where a n a n 1 a n and 2 a n a n then is convergent and the constant 4 being best possible equality occurs if and only if a n 0 for all n in this paper we give a result that extends copson s result to inequalities of the form where mx n p n l x n l q n x n n 0 1 the validity of such an inequality and the best possible value of the constant k are determined in terms of the analogue of the titchmarsh weyl m function for the difference equation mx n w n x n n 0 1

graded manifolds supermanifolds and infinite dimensional grassmann algebras a new approach to superdifferentiable functions of grassmann variables is developed which avoids ambiguities in odd derivatives this is used to give an improved definition of supermanifold over a finite dimensional grassmann algebra a natural embedding of super manifolds over grassmann algebras with increasing number l of generators is developed and thus a limit asl tends to infinity is possible a correspondence between graded manifolds and supermanifolds is constructed extending results of 5 and 8

on schroder s families of root finding methods schroder s methods of the first and second kind for solving a nonlinear equation f x 0 originally derived in 1870 are of great importance in the theory and practice of iteration processes they were rediscovered several times and expressed in different forms during the last 130 years it was proved in the paper of petkovic and herceg 1999 7 that even seven families of iteration methods for solving nonlinear equations are mutually equivalent in this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in schroder s generalized method of the second kind presented in 1870 in the continuation we consider smale s open problem from 1994 about possible link between schroder s methods of the first and second kind and state the link in a simple way

enlargeable graded lie algebras of supersymmetry a criterion to enlarge infinite dimensional lie algebras to analytic lie groups is used here for the extension of graded lie algebras of physical interest to super lie groups