interpolatory quadrature formulas on the unit circle for chebyshev weight functions in this paper interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands a comparison is made with szego quadrature formulas finally an application to the interval 1 1 is also carried out

discrete integrable systems and deformations of associative algebras interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed theory of deformations for associative algebras is presented closed left ideal generated by the elements representing the multiplication table plays a central role in this theory deformations of the structure constants are generated by the deformation driving algebra and governed by the central system of equations it is demonstrated that many discrete equations such as discrete boussinesq equation discrete wdvv equation discrete schwarzian kp and bkp equations discrete hirota miwa equations for kp and bkp hierarchies are particular realizations of the central system an interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful an interpretation of the menelaus relation discrete schwarzian kp equation discrete hirota miwa equation for kp hierarchy consistency around the cube as the associativity conditions and the concept of gauge equivalence for instance between the menelaus and kp configurations are particular examples

critical properties of random site percolation in two and three dimensions a monte carlo study abstract we measure the critical exponents of two dimensional and three dimensional random site percolation and find excellent agreement with nienhuis exact results two dimensions and good agreement with other numerical work three dimensions we also measure the correlation length amplitude ratio and the mean cluster size amplitude ratio in two and three dimensions we report values of 4 0 0 5 2 d 2 0 0 5 3 d and 75 40 26 2 d and 8 3 2 3 d for the correlation length and the mean cluster size amplitude ratios respectively a direct measurement is made of the fractal dimension d f at the critical point p c 0 50 of the triangular lattice d f is 1 90 0 01 while at p c 0 3117 for the simple cubic lattice d f is 2 50 0 02

a nested iterative scheme for indefinite linear systems in particulate flows abstract high fidelity large scale direct numerical simulation of particulate flows is of great value in a variety of industrial applications it is computationally intensive as it combines time integration solving nonlinear algebraic equations and the associated linear systems the finite element discretization of the coupled system of pdes on an unstructured grid using an arbitrary lagrangian eulerian moving mesh technique leads to very large nonlinear systems that are linearized by a version of newton s method the linear algebraic systems jacobians are sparse nonsymmetric and indefinite for which standard linear system solvers based on krylov subspace methods generally fail to converge without appropriate preconditioners the failure of krylov methods in production codes is currently being addressed by reducing the size of the time step this however leads to a very long simulation time and therefore is not always a viable approach in this study we design a hybrid inner outer iterative scheme for solving these indefinite systems which proves to be both efficient robust and ideally suited for parallel computing platforms even with appropriate large time steps comparisons with krylov subspace methods show the superiority of our proposed class of nested iterative schemes which is also scalable with respect to mesh size and insensitive to changes in properties of the fluid particles system

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

exact enumeration approach to fractal properties of the percolation backbone and 1 σ expansion an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold the authors use this approach to calculate exactly the first low density expansion of lbb p for arbitrary system dimensionality d where lbb p is the mean of backbone bonds and p is the bond occupation probability standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d these disagree with the sierpinski gasket model of the backbone they also calculate the first low density expansions of lmin p and lred p which are respectively the mean number of bonds in the minimum path between i and j and the mean number of singly connected red bonds

integrable discrete differential geometry of plated membranes in equilibrium recently it has been established that a well known system of classical shell theory descriptive of membranes in equilibrium is in fact amenable to the techniques of soliton theory here it is d

on the numerical resolution of the generalized airfoil equation with possio kernel in this paper we consider the so called generalized airfoil singular integral equation with a smooth input function and present ad hoc quadrature rules to compute efficiently the elements of collocation and galerkin matrices

quantization of three wave equations the subject of this paper is the revision of discretization and quantization of two similar classical integrable hamiltonian systems in three dimensional spacetime the standard three wave equations and the modified three wave equations the darboux manakov zakharov system in two distinguished gauges the quantized systems in discrete spacetime may be understood as the regularized integrable quantum field theories integrability of the theories and in particular the quantum tetrahedron equations for vertex operators follow from the quantum auxiliary linear problems the principal object of the lattice field theories is the heisenberg discrete time evolution operator constructed with the help of vertex operators

fundamental transformations for quadrilateral lattices first potentials and τ functions symmetric and pseudo egorov reductions we find the behaviour of first potentials and functions for quadrilateral lattices under vectorial fundamental transformations we also give those transformations which preserve the symmetric and pseudo egorov reductions

from integrable nets to integrable lattices inspired by the results of jonas einsenhart demoulin and bianchi on the permutability property of classical geometrical transformations of conjugate nets and its reductions of pseudo orthogonal pseudo symmetric and pseudo egorov types dressing transformations of the n component kp hierarchy described within the grassmannian are used to generate quadrilateral lattices and its corresponding reductions as a byproduct we get the corresponding discrete dressing transformations in particular we characterize the vectorial fundamental discrete transformations preserving the symmetric lattice

interpolary product integration in the presence of singularities l 2 theory we study interpolatory product integration rules based on gauss radau and lobatto points with respect to a generalized jacobi weight function as applied to the integration of functions with singularities we give sufficient conditions for convergence of such rules for both endpoint and interior singularities

invasion percolation new algorithms and universality classes employing highly efficient algorithms for simulating invasion percolation ip whose execution time scales as o mlog m or better for a cluster of m sites and for determining the backbone of the cluster we obtain precise estimates for the fractal dimensions of the sample spanning cluster the backbone and the minimal path in order to identify the universality classes of four different ip processes site and bond ip with and without trapping in two dimensions ip is characterized by two universality classes one each for ip without trapping and site and bond ip with trapping in a three dimensional site ip with and without trapping is in the universality class of random percolation while bond ip with trapping is in a distinct universality class which may be the same as that of optimal paths in strongly disordered media

theory of self avoiding walks on percolation fractals a phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self avoiding walks saw is used to derive a new expression for the flory exponent describing the average radius of gyration of saws on fractals we focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of saws the mean field version of our new formula corresponds to the aharony and harris expression who used the standard flory approach for its derivation

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

on the geometry of an integrable 2 1 dimensional sine gordon system it is recorded that darboux9s method of linking the classical lame system governing triply orthogonal systems of surfaces with an integrable 2 1 dimensional sine gordon equation may be extended and applied to the integrable two component generalization of the latter introduced by konopelchenko and rogers thus in a reinterpretation this 2 1 dimensional sine gordon system is shown to define particular integrable motions of surfaces

charged free fermions vertex operators and the classical theory of conjugate nets we show that the quantum field theoretical formulation of the function theory has a geometrical interpretation within the classical transformation theory of conjugate nets in particular we prove that i the partial charge transformations preserving the neutral sector are laplace transformations ii the basic vertex operators are levy and adjoint levy transformations and iii the diagonal soliton vertex operators generate fundamental transformations we also show that the bilinear identity for the multicomponent kadomtsev petviashvili hierarchy becomes through a generalized miwa map a bilinear identity for the multidimensional quadrilateral lattice equations

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

three dimensional integrable lattices in euclidean spaces conjugacy and orthogonality it is shown that the discrete darboux system descriptive of conjugate lattices in euclidean spaces admits constraints on the adjoint eigenfunctions which may be interpreted as discrete orthogonality conditions on the lattices thus it turns out that the elementary quadrilaterals of orthogonal lattices are cyclic orthogonal lattices on lines planes and spheres are discussed and the underlying integrable systems in one two and three dimensions are derived explicitly a discrete analogue of bianchi s ribaucour transformation is set down and particular orthogonal lattices are given as a by product discrete dini surfaces are obtained

asymptotic lattices and their integrable reductions i the bianchi ernst and the fubini ragazzi lattices we review recent results on asymptotic lattices and their integrable reductions we present the theory of general asymptotic lattices in 3 together with the corresponding theory of their darboux type transformations then we find a novel permutability theorem for bianchi surfaces which can be reinterpreted as a discrete version of the bianchi ernst system and coincides with an equation recently introduced by schief schief w k 2001 stud appl math 106 85 137 using the well known connection between the bianchi and ernst systems we also propose the discrete analogue of the ernst system finally we present the theory of the discrete analogues of isothermally asymptotic fubini ragazzi nets together with their transformations

rates of convergence of multipoint rational approximants and quadrature formulas on the unit circle abstract in this paper multipoint rational approximants to the riesz herglotz transform of a borel measure supported on are considered we give estimates for the rate of convergence of these approximants these estimates are obtained from the asymptotic behaviour of a sequence of rational functions orthogonal with respect to and having prescribed poles 1 k where k 1 is a sequence of complex numbers compactly contained in the open unit disk as an application we also give the rate of convergence of rational szego quadrature formulas for integrals with respect to

generalized isothermic lattices we study multi dimensional quadrilateral lattices satisfying simultaneously two integrable constraints a quadratic constraint and the projective moutard constraint when the lattice is two dimensional and the quadric under consideration is the mobius sphere one obtains after the stereographic projection the discrete isothermic surfaces defined by bobenko and pinkall by an algebraic constraint imposed on the complex cross ratio of the circular lattice we derive the analogous condition for our generalized isothermic lattices using steiner s projective structure of conics and we present basic geometric constructions which encode integrability of the lattice in particular we introduce the darboux transformation of the generalized isothermic lattice and we derive the corresponding bianchi permutability principle finally we study two dimensional generalized isothermic lattices in particular geometry of their initial boundary value problem

reductions of the multidimensional quadrilateral lattice the multidimensional circular lattice we apply a recently introduced 21 15 reduction method based on the dressing to construct a large class of integrable reductions of the equations characterizing the multidimensional quadrilateral lattice an n dimensional lattice in m n m whose elementary quadrilaterals are planar and whose continuous limit describes submanifolds parametrized by conjugate lines 11 we also show that generically in the limit of the small lattice parameter half of these reductions lead to the darboux equations for symmetric fields and the second half lead to the generalized lame equations describing n dimensional submanifolds of m parametrized by conjugate orthogonal systems of coordinates we finally show that a distinguished example of the second class of reductions corresponds to the multidimensional circular lattice an n dimensional lattice in m n m whose elementary quadrilaterals are inscribed in circles 7

rational quadrature formulae on the unit circle with arbitrary poles interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered the poles are prescribed under the only restriction of not lying on the unit circle a computable upper bound of the error is obtained which is valid for any choice of poles arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle a number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound also a comparison is made with other error bounds appearing in the literature

characterizing the measures on the unit circle with exact quadrature formulas in the space of polynomials in the present paper we characterize the measures on the unit circle for which there exists a quadrature formula with a fixed number of nodes and weights and such that it exactly integrates all the polynomials with complex coefficients as an application we obtain quadrature rules for polynomial modifications of the bernstein measures on 1 1 having a fixed number of nodes and quadrature coefficients and such that they exactly integrate all the polynomials with real coefficients

darboux transformations for a 6 point scheme we introduce the jonas transformation a transformation of the darboux type for the general second order differential equation in two independent variables we present a discrete version of the transformation for a 6 point difference scheme the scheme is appropriate for the solution of a hyperbolic type initial boundary value problem we discuss several reductions and specifications of the transformations

geometric discretization of the koenigs nets we introduce the koenigs lattice which is a new integrable reduction of the quadrilateral lattice discrete conjugate net and provides natural integrable discrete analog of the koenigs net we construct the darboux type transformation of the koenigs lattice and we show permutability of superpositions of such transformations thus proving integrability of the koenigs lattice we also investigate the geometry of the discrete koenigs transformation in particular we characterize the koenigs transformation in terms of an involution determined by a congruence conjugate to the lattice

product integration based on hermite feje r interpolation abstract the use of generalized hermite fejer interpolation polynomials for product integration is studied algorithms for implementing this method are developed and convergence results are proved comparisons are made between this new approach and the classical approach based on lagrange interpolation polynomials it turns out that integration rules based on hermite fejer interpolation converge in more cases but in general more slowly than those based on lagrange interpolation when the latter converge

integrable discretization of hodograph type systems hyperelliptic integrals and whitham equations based on the well established theory of discrete conjugate nets in discrete differential geometry we propose and examine discrete analogues of important objects and notions in the theory of semi h

transformations of quadrilateral lattices motivated by the classical studies on transformations of conjugate nets we develop the general geometric theory of transformations of their discrete analogs the multidimensional quadrilateral lattices i e lattices x zn rm n m whose elementary quadrilaterals are planar our investigation is based on the discrete analog of the theory of the rectilinear congruences which we also present in detail we study in particular the discrete analogs of the laplace combescure levy radial and fundamental transformations and their interrelations the composition of these transformations and their permutability is also investigated from a geometric point of view the deep connections between transformations and discretizations is also investigated for quadrilateral lattices we finally interpret these results within the formalism

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

scaling relation for the critical exponents of the backbone of percolation clusters the author suggests that beta b the critical exponent of the backbone of percolation clusters is given by beta b 1 2 gamma 5 beta 1 where gamma and beta are susceptibility and order parameter exponents of percolation respectively the proposed relation agrees with the epsilon expansion of beta b given by harris and lubensky and provides accurate estimates of beta b at all d 1 this relation may suggest that the backbone exponents are not independent of other percolation exponents the author also suggests a more general relation for the order parameter exponent of m connected clusters

a connection between quadrature formulas on the unit circle and the interval 1 1 abstract we establish a relation between gauss quadrature formulas on the interval 1 1 that approximate integrals of the form i f 1 1 f x x d x and szeg quadrature formulas on the unit circle of the complex plane that approximate integrals of the form i f f e i d the weight x is positive on 1 1 while the weight is positive on it is shown that if cos sin then there is an intimate relation between the gauss and szeg quadrature formulas moreover as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to x and orthogonal szeg polynomials with respect to inclusion of gauss lobatto and gauss radau formulas is natural

darboux transformations for linear operators on two dimensional regular lattices darboux transformations for linear operators on regular two dimensional lattices are reviewed the six point scheme is considered as the master linear problem whose various specifications reductions and sublattice combinations lead to other linear operators together with the corresponding darboux transformations the second part of the review deals with multidimensional aspects of basic reductions of the four point scheme as well as the three point scheme

discrete line complexes and integrable evolution of minors based on the classical plcker correspondence we present algebraic and geometric properties of discrete integrable line complexes in cp3 algebraically these are encoded in a discrete integrable s

numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures numerical methods for strongly oscillatory and singular functions are given in this paper beside a summary of standard methods and product integration rules we consider a class of complex integration methods several applications of quadrature processes in problems in telecom munications and physics are also presented 1998 elsevier science ltd all rights reserved integration oscillatory kernel singular kernel orthogonal polynomials product rules gaussian quaclratures error function bessel functions legendre functions

lattice and q difference darboux zakharov manakov systems via delta dressing method a general scheme is proposed for the introduction of lattice and q difference variables to integrable hierarchies in the frame of the delta dressing method using this scheme lattice and q difference darboux zakharov manakov systems of equations are derived darboux backlund and combescure transformations and exact solutions for these systems are studied

fractal properties of the percolating backbone in three dimensions a recursive algorithm to determine the backbone of the two dimensional percolating cluster is adjusted to measure the fractal properties of the backbone of the three dimensional percolating cluster the specific properties measured are the fractal dimensionality of the backbone dbbf 1 75 or 0 04 the chemical dimensionality dt 1 26 or 0 03 and the minimum path dimensionality dmin 1 39 or 0 03

product integration over infinite intervals i rules based on the zeros of hermite polynomials the paper discusses both theoretical properties and practical implementation of product integration rules of the form n j k x f x dx wnlj xnj o i l where f is continuous k is absolutely integrable the nodes xn are roots of the hermite polynomials hn x and the weights wn are chosen so that the rule is exact if f is any polynomial of degree n convergence of the rule to the exact integral as n oo is proved for a wide class of functions f and k including singular or oscillatory functions k and rates of convergence are estimated the rules are shown to have the property of asymptotic positivity and as a consequence exhibit good numerical stability numerical calculations for some practical cases are presented which show the method to be computationally effective for integrands including highly oscillatory ones that decay suitably at infinity applications of the method to integration over 0 oo are also discussed

the curvature line parametrization from circular nets on a surface we study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets circular nets and planar quadrilateral nets respectively with edges of order both smooth and discrete geometries are described by integrable systems it is shown that one can obtain an order 2 approximation globally with points of the discrete nets on the smooth surface a new simple geometric construction of principal directions of smooth surfaces is given we study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets circular nets and planar quadrilateral nets respectively with edges of order both smooth and discrete geometries are described by integrable systems it is shown that one can obtain an order 2 approximation globally with points of the discrete nets on the smooth surface a new simple geometric construction of principal directions of smooth surfaces is given

error bounds for rational quadrature formulae of analytic functions we study the error of rational quadrature rules when functions which are analytic on a neighborhood of the set of integration are considered a computable upper bound of the error is presented which is valid for a broad range of rational quadrature formulae and a comparison is made with the exact error for a number of numerical examples

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given

on the unification of classical and novel integrable surfaces ii difference geometry a novel class of discrete integrable surfaces is recorded this class of discrete o surfaces is shown to include discrete analogues of classical surfaces such as isothermic linear weingarten guichard and petot surfaces moreover natural discrete analogues of the gaussian and mean curvatures for surfaces parametrized in terms of curvature coordinates are used to define surfaces of constant discrete gaussian and mean curvatures and discrete minimal surfaces remarkably these turn out to be prototypical examples of discrete o surfaces it is demonstrated that the construction of a backlund transformation for discrete o surfaces leads in a natural manner to an associated parameter dependent linear representation canonical discretizations of the classical pseudosphere and breather pseudospherical surfaces are generated connections with pioneering work by bobenko and pinkall are established