quantized knizhnik zamolodchikov equations quantum yang baxter equation and difference equations for q hypergeometric functions thesl2 quantized knizhnik zamolodchikov equations are solved inq hypergeometric functions new difference equations are derived for generalq hypergeometric functions the equations are given in terms of quantum yang baxter matrices and have the form similar to quantum knizhnik zamolodchikov equations for quantum affine algebras introduced by frenkel and reshetikhin

tightness and distinguished frechet spaces in this chapter we apply the concept of tightness to study distinguished frechet spaces we show that a frechet space is distinguished if and only if its strong dual has countable tightness this approach to studying distinguished frechet spaces leads to a rich supply of df spaces whose weak duals are quasi suslin but not k analytic the small cardinals mathfrak b and mathfrak d will be used to improve the analysis of kothe s echelon nondistinguished frechet space 1 a

fock space representations of affine lie algebras and integral representations in the wess zumino witten models fock space representations of affine lie algebras are studied explicit forms of correction terms adding to the currentsf i z are determined it is proved that the sugawara energy momentum tensor on the fock spaces is quadratic in free bosons furthermore screening operators are constructed this implies the existence of generalized hypergeometric integrals satisfying the knizhnik zamolodchikov equation

weierstrass formula and zero finding methods classical weierstrass formula 29 has been often the subject of investigation of many authors in this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions we are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros conditions for the safe convergence of weierstrass method depending only on initial approximations are given in particular we study polynomials with interval coefficients using an interval version of weierstrass method enclosures in the form of disks for the complex valued set containing all zeros of a polynomial with varying coefficients are obtained we also present weierstrass like algorithm for approximating simultaneously all zeros of a class of analytic functions in a given closed region to demonstrate the proposed algorithms three numerical examples are included

a q deformation of wakimoto modules primary fields and screening operators theq vertex operators of frenkel and reshetikhin are studied by means of aq deformation of the wakimoto module for the quantum affine algebrau q widehat mathfrak s mathfrak l 2 at an arbitrary levelk 0 2 a fock module version of theq deformed primary field of spinj is introduced as well as the screening operators which anti commute with the action ofu q widehat mathfrak s mathfrak l 2 up to a total difference of a field a proof of the intertwining property is given for theq vertex operators corresponding to the primary fields of spinj 1 2z 0 a sample calculation of the correlation function is also given

on approximate zeros and rootfinding algorithms for a complex polynomial on donne des criteres pour qu un nombre complexe soit une racine approchee d un polynome f par la methode de newton ou d euler d ordre k on construit des familles d algorithmes surement convergents

locating multiple zeros interactively abstract we present a globally convergent algorithm which has been implemented interactively for the total or partial factorisation of a polynomial initially a region is found containing all the zeros of a polynomial a parallel search algorithm is then used to locate successively smaller regions containing zeros this continues until it is possible to estimate the multiplicities of the zeros reliably when a rapidly converging iteration function if takes over at this point the user using an interactive terminal can elect to investigate further all of the zeros or only those in certain regions the if is selected from a class of such functions which exhibit certain desirable mathematical and computational properties a synopsis of computational results is given

weight of precompact subsets and tightness pfister 1976 and cascales and orihuela 1986 proved that precompact sets in df and lm spaces have countable weight i e are metrizable improvements by valdivia 1982 cascales and orihuela 1987 and kakol and saxon preprint have varying methods of proof for these and other improvements a refined method of upper semi continuous compact valued maps applied to uniform spaces will suffice at the same time this method allows us to dramatically improve kaplansky s theorem that the weak topology of metrizable spaces has countable tightness extending it to include all lm spaces and all quasi barrelled df spaces both in the weak and original topologies one key is showing that for a large class g including all df and lm spaces countable tightness of the weak topology of e in g is equivalent to realcompactness of the weak topology of the dual of e

difference equations for correlation functions of belavin s ℤn symmetric model with boundary reflection belavin s n symmetric elliptic model with boundary reflection is considered on the basis of the boundary ctm bootstrap we find non diagonal k matrices for n 2 that satisfy the reflection equation boundary yang baxter equation and also find non diagonal boltzmann weights for the a 1 n 1 face model even for n 2 we derive difference equations of the quantum knizhnik zamolodchikov type for correlation functions of the boundary model the boundary spontaneous polarization is obtained by solving the simplest difference equations in the case of the free boundary condition the resulting quantity is the square of the spontaneous polarization for the bulk n symmetric model up to a phase factor

on the convergence order of accelerated root iterations a gauss seidel procedure for accelerating the convergence of the generalized method of the root iterations type of the k 2 th order k n for finding polynomial complex zeros given in 7 is considered in this paper it is shown that ther order of convergence of the accelerated method is at leastk 1 n k where n k 1 is the unique positive root of the equation n k 1 0 andn is the degree of the polynomial the examples of algebraic equations in ordinary and circular arithmetic are given

point estimation of simultaneous methods for solving polynomial equations a survey ii the construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root finding algorithm is one of the most important problems in solving nonlinear equations smale s point estimation theory from 1981 was a great advance in this topic it treats convergence conditions and the domain of convergence in solving an equation f z 0 using only the information of f at the initial point z 0 the study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult even in the case of algebraic polynomials in the light of smale s point estimation theory an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials we state new improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations ehrlich aberth s method ehrlich aberth s method with newton s correction borsch supan s method with weierstrass correction and halley like or wang zheng method the introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods but also explicitly gives their order of convergence the stated initial conditions are of significant practical importance since they are computationally verifiable they depend only on the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

computer methodologies for comparison of computational efficiency of simultaneous methods for finding polynomial zeros abstract the study and analysis of iterative methods for the determination of zeros of algebraic polynomials the subject of this paper is one of the most actual tasks in the theory and practice our main goal is to examine and compare the quality of most frequently used methods for simultaneously finding all zeros of a polynomial by combining numerical experiments theoretical models and empirical cpu time methodology special attention is devoted to empirical average case model assuming that average cpu times are calculated performing the experiment i on several different computers ii running each method through many loops for a fixed example iii testing reasonably large number of polynomials of different structure body and iv the use of almost arbitrary initial approximations the empirical model ranks the tested methods on the base of the consumed cpu times needed for the fulfillment of the required accuracy of approximations under the same initial conditions our aim is to select a group of the most powerful simultaneous methods with globally convergent performance convergence behavior of two most powerful methods is illustrated by trajectories in the complex plane that consist of the point approximations starting with initial points and going towards the endpoints where the iterative process stops due to the termination criterion

lf spaces quasi baire spaces and the strongest locally convex topology on e coarser than r n 1 2 if e 0 e and is the finest locally convex n l topology on e which induces on each e a topology coarser than then e is said to be the inductive limit of the sequence e z ithe family e z 1 is a definin9 sequence for the inductive limit space e z note that we are using a narrow definition of the notion of an inductive limit that befits our needs if each e r is a fr6chet banach space e e 1 and z is hausdorff then the inductive limit e z is said to be an lf space lb space and we write

traub gander s family for the simultaneous determination of multiple zeros of polynomials abstract by combining traub gander s family of third order for finding a multiple zero and suitable corrective approximations od schroder s and halley s type a new family of iterative methods for the simultaneous approximation of multiple zeros of algebraic polynomials is proposed taking various forms of a function involved in the iterative formula a number of different simultaneous methods can be obtained it is proved that the order of convergence is 4 5 or 6 depending of the type of employed corrective approximations two numerical examples are given to demonstrate the convergence properties of the proposed family of simultaneous methods displayed trajectories of the sequences of approximations point to global characteristics of the proposed family of iterative methods

jackson integrals of jordan pochhammer type and quantum knizhnik zamolodchikov equations we show that theq difference systems satisfied by jackson integrals of jordan pochhammer type give a class of the quantum knizhnik zamolodchikov equation for u q widehat mathfrak s mathfrak l 2 in the sense of frenkel and reshetikhin

quantum knizhnik zamolodchikov equation for uq sln and integral formula an integral formula for solutions to the level zero quantum knizhnik zamolodchikov equation associated with the vector representation of uq sln is presented this formula gives a generalization of both our previous work for uq sln and smirnov s formula for form factors of the su n chiral gross neveu model

approximate zeros of quadratically convergent algorithms smale s condition for a point to be an approximate zero of a func tion for newton s method is extended to the general quadratically convergent iterative algorithm it is shown in which way the bound in the condition is affected by the characteristics of the algorithm this puts the original condition of smale for newton s method in a more general perspective the results are also discussed in the light of numerical evidence

quantum algebra structure of certain jackson integrals theq difference system satisfied by jackson integrals with a configuration ofa type root system is studied we explicitly construct some linear combination of jackson integrals which satisfies the quantum knizhnik zamolodchikov equation for the 2 point correlation function ofq vertex operators introduced by frenkel and reshetikhin for the quantum affine algebra u q widehat mathfrak s mathfrak l 2 the expression of integrands for then point case is conjectured and a set of linear relations for the corresponding jackson integrals is proved

improved halley like methods for the inclusion of polynomial zeros improved iterative methods of halley s type for the simultaneous inclusion of all simple complex zeros of a polynomial are proposed the presented convergence analysis which uses the concept of the r order of convergence of mutually dependent sequences shows that the convergence rate of the basic fourth order method is increased to 5 and 6 using newton s and halley s corrections respectively the proposed inclusion methods possess a high computational efficiency since the increase of the convergence is attained without additional calculations this advantage together with very fast convergence make that the presented methods are ranking among the most powerful inclusion methods for polynomial zeros in order to demonstrate convergence properties of the proposed methods two numerical examples are given

an application of aomoto gelfand hypergeometric functions to the su n knizhnik zamolodchikov equation solutions to the knizhnik zamolodchikov equation for verma modules of the lie algebra mathfrak s mathfrak l n 1 mathbb c are explicitly given by certain integrals called aomoto gelfand hypergeometric functions

on an iterative method for simultaneous inclusion of polynomial complex zeros abstract starting from disjoint discs which contain polynomial complex zeros the iterative interval method of the third order for the simultaneous finding inclusive discs for complex zeros is formulated the lagrangean interpolation formula and complex circular arithmetic are used the convergence theorem and the conditions for convergence are considered the proposed method has been applied for solving an algebraic equation

smirnov s integrals and the quantum knizhnik zamolodchikov equation of level 0 we study the quantum knizhnik zamolodchikov equation 1984 of level 0 associated with the spin 1 2 representation of uq sl2 we find an integral formula for solutions in the case of an arbitrary total spin and mod q mod 1 in the formula different solutions can be obtained by taking different integral kernels with the cycle of integration being fixed

annihilation poles of a smirnov type integral formula for solutions to the quantum knizhnik zamolodchikov equation we consider the recently obtained integral representation of the quantum knizhnik zamolodchikov equation of level 0 we obtain the condition for the integral kernel such that these solutions satisfy three axioms for the form factor in the manner of smirnov we discuss the relation between this integral representation and the form factor of the xxz spin chain

the theory of smale s point estimation and its applications abstract the main result of this paper is that we exact smale s point estimation theory i e without assuming k p z 1 p k z k k 2 being bounded by the point estimation convergence theorem of the ne method is set up by making use of the majorizing method the proof of the theorem is simple and precise while the required point estimation conditions are weaker than all those of known point estimation convergence theorems another result of this paper is an application of the above new theory to the durand kerner method we compare the point estimation conditions for the durand kerner method with other known point estimation conditions numerical results show that our results have evident advantages

form factors of the xxz model and the affine quantum group symmetry we present new expressions of form factors of the xxz model which satisfy smirnov s three axioms these new form factors are obtained by applying the affine quantum group to the known ones obtained in our previous works we also find the relations among all the new and known form factors i e all other form factors are obtained by applying to a singlet form factor

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

askey wilson polynomials by means of a q selberg type integral abstract an integral representation of the askey wilson polynomials is presented in terms of a q selberg type integral our motivation consists in the study of q selberg type integrals from the viewpoint of de rham theory or holonomic systems

quantum screening operators and canonical q de rham cocycles in this paper a close connection is established between certain cohomology spaces of representations of the quantum affine algebra and a twisted q de rham jackson aomoto cohomology of configuration spaces using the quantum screening operators

bounded tightness for locally convex spaces and spaces c x abstract we show that metrizability and bounded tightness are actually equivalent for a large class g of locally convex spaces including lf spaces df spaces the space of distributions d etc a consequence of this fact is that for x g the bounded tightness for the weak topology of x is equivalent to the following one x is linearly homeomorphic to a subspace of r n this nicely supplements very recent results of cascales and raja moreover we show that a metric space x is separable if the space cp x has bounded tightness

vertex representations of quantum affine algebras we construct vertex representations of quantum affine algebras of ade type which were first introduced in greater generality by drinfeld and jimbo the limiting special case of our construction is the untwisted vertex representation of affine lie algebras of frenkel kac and segal our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations in the case of the quantum affine algebra of type a we introduce vertex operators corresponding to all the roots and determine their commutation relations this provides an analogue of a chevalley basis of the affine lie algebra unk n in the basic representation

on countable bounded tightness for spaces cp x abstract it is well known that the space cp 0 1 has countable tightness but it is not frechet urysohn let x be a cech complete topological space we prove that the space cp x of continuous real valued functions on x endowed with the pointwise topology is frechet urysohn if and only if cp x has countable bounded tightness i e for every subset a of cp x and every x in the closure of a in cp x there exists a countable and bounding subset of a whose closure contains x we study also the problem when the weak topology of a locally convex space has countable bounded tightness additional results in this direction are provided

on the convergence of wang zheng s method the choice of initial conditions ensuring safe convergence of the implemented iterative method is one of the most important problems in solving polynomial equations these conditions should depend only on the coefficients of a given polynomial p and initial approximations to the zeros of p in this paper we state initial conditions with the described properties for the wang zheng method for the simultaneous approximation of all zeros of p the safe convergence and the fourth order convergence of this method are proved

on halley like algorithms for simultaneous approximation of polynomial complex zeros let p z be a polynomial of degree n with n distinct zeros iterative methods for the simultaneous inclusion of k leqq n simple or multiple zeros are given the iterative formulas presented can be regarded as an interval version of halley s iterative formula for multiple zeros realized in circular arithmetic the convergence analysis is done in the presence of multiplicity the convergence order is three for k n and four if k n applying the gauss seidel approach to the basic iterative formula the single step method with accelerated convergence is stated and the lower bound of the r order of convergence of this method is obtained the dependence of convergence rate of the basic methods on the rounding errors is also studied some numerical examples illustrate the theoretical results