chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

a divergence free multidomain spectral solver of the navier stokes equations in geometries of high aspect ratio in order to solve with high accuracy the incompressible navier stokes equations in geometries of high aspect ratio one has developed a spectral multidomain algorithm well adapted to the parallel computing in cases of 2d problems a chebyshev collocation method and an extended influence matrix technique are used in each subdomain to solve the generalized stokes problem which results from the discretization in time the continuity conditions needed at the interfaces of each subdomain are computed by using again an influence matrix the setup of which ensures all the necessary compatibility conditions especially the incompressibility this work is also described for 3d problems with one homogeneous direction a study of accuracy versus the number of subdomains is presented as well as an example of an application concerned with rayleigh benard convection in a cavity of large aspect ratio

exact integration of the unsteady incompressible navier stokes equations gauge criteria and applications an exact first integral of the full unsteady incompressible navier stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with maxwell s theory subsequent to this gauge freedoms are explored showing that when used astutely they lead to a favourable reduction in the complexity of the associated equation set and number of unknowns following which the inviscid limit case is discussed finally it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self adjoint form use of the new formulation is demonstrated for different gauge variants of the first integral as the starting point through the solution of a hierarchy of classical three dimensional flow problems two of which are tractable analytically the third being solved numerically in all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature concurrently the prescription of appropriate commonly occurring physical and necessary auxiliary boundary conditions incorporating for completeness the derivation of a first integral of the dynamic boundary condition at a free surface is established together with how the general approach can be advantageously reformulated for application in solving unsteady flow problems with periodic boundaries

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

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

solving the quintic by iteration equations that can be solved using iterated rational maps are characterized an equation is computable if and only if its galois group is within a5 of solvable we give explicitly a new solution to the quintic polynomial in which the transcendental inversion of the icosahedral map due to hermite and kronecker is replaced by a purely iterative algorithm the algorithm requires a rational map with icosahedral symmetries we show all rational maps with given symmetries can be described using the classical theory of invariant polynomials

regular article on a class of pade finite volume methods a class of pade finite volume methods providing an improved spectral resolution is presented and compared with well known methods the formulation is based on the sliding averages of the variables and allows the computation of derivatives of all orders using the fourier analysis these methods are examined with respect to i order of accuracy ii spectral resolution iii boundary conditions and iv stability

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

attracting cycles for the relaxed newton s method we study the relaxed newton s method applied to polynomials in particular we give a technique such that for any n 2 we may construct a polynomial so that when the method is applied to a polynomial the resulting rational function has an attracting cycle of period n we show that when we use the method to extract radicals the set consisting of the points at which the method fails to converge to the roots of the polynomial p z z m c this set includes the julia set has zero lebesgue measure consequently iterate sequences under the relaxed newton s method converge to the roots of the preceding polynomial with probability one

new identities in the catalan triangle abstract in this paper we prove new identities in the catalan triangle whose n p entry is defined by b n p p n 2 n n p n p n p n in fact we show some new identities involving the well known catalan numbers and specially the identity p 1 i b n p b n n p i n 2 p i n 1 c n 2 n 1 i 1 i n that appears in a problem related with the dynamical behavior of a family of iterative methods applied to quadratic polynomials

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

vorticity vector potential method for 3d viscous incompressible flows in time dependent curvilinear coordinates e and liu j comput phys 138 1997 57 82 put forward a finite difference method for 3d viscous incompressible flows in the vorticity vector potential formulation on non staggered grids in this paper we will extend this method to the case of flows in the presence of a deformable surface by use of two kinds of surface differential operators the implementation of boundary conditions on a plane is generalized to a curved smooth surface with given velocity distribution whether this be an inflow outflow interface or a curved wall to deal with the irregular and varying physical domain time dependent curvilinear coordinates are constructed and the corresponding tensor analysis is adopted in deriving the component form of the governing equations therefore the equations can be discretized and solved in a regular and fixed parametric domain numerical results are presented for a 3d lid driven cavity with a deforming surface and a 3d duct flow with a deforming boundary a new way to validate numerical simulations is proposed based on an expression for the rate of strain tensor on a deformable surface

moduli space of cubic newton maps abstract in this article we study the topology and bifurcations of the moduli space m 3 of cubic newton maps it s a subspace of the moduli space of cubic rational maps carrying the riemann orbifold structure c 2 3 we prove two results the boundary of the unique unbounded hyperbolic component is a jordan arc and the boundaries of all other hyperbolic components are jordan curves the head s angle map is surjective and monotone the fibers of this map are characterized completely the first result is a moduli space analogue of the first author s dynamical regularity theorem 37 the second result confirms a conjecture of tan lei

newton method for symmetric quartic polynomial we investigate the parameter plane of the newton s method applied to the family of quartic polynomials p a b z z 4 a z 3 b z 2 a z 1 where a and b are real parameters we divide the parameter plane a b r 2 into twelve open and connected regions where p p and p have simple roots in each of these regions we focus on the study of the newton s operator acting on the riemann sphere

families of rational maps and iterative root finding algorithms in this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root finding algorithms we answer a question of smale s by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more we settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics and we give a classification of all such algorithms in the context of conformal dynamics our main result is the following a stable algebraic family of rational maps is either trivial all its members are conjugate by mobius transformations or affine its members are obtained as quotients of iterated addition on a family of complex tori our classification of generally convergent algorithms follows easily from this result as another consequence of rigidity we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices this implies that bounded analytic functions nearly separate points on the moduli space of a rational map

the moduli space of polynomial maps and their fixed point multipliers abstract we consider the family mp d of affine conjugacy classes of polynomial maps of one complex variable with degree d 2 and study the map d mp d d c d s d which maps each f mp d to the set of fixed point multipliers of f we show that the local fiber structure of the map d around d is completely determined by certain two sets i and k which are subsets of the power set of 1 2 d moreover for any d we give an algorithm for counting the number of elements of each fiber d 1 only by using i and k it can be carried out in finitely many steps and often by hand

approximation of non archimedean lyapunov exponents and applications over global fields let k be an algebraically closed field of characteristic 0 that is complete with respect to a non archimedean absolute value we establish a locally uniform approximation formula of the lyapunov exponent of a rational map f of mathbb p 1 of degree d 1 over k in terms of the multipliers of n periodic points of f with an explicit control in terms of n f and k as an immediate consequence we obtain an estimate for the blow up of the lyapunov exponent near a pole in one dimensional families of rational maps over k combined with our former archimedean version this non archimedean quantitative approximation allows us to show a quantified version of silverman s and ingram s recent comparison between the critical height and any ample height on the moduli space mathcal m d bar mathbb q two improvements of mcmullen s finiteness of the multiplier maps reduction to multipliers of cycles of exact given period and an effective bound from below on the period a characterization of non affine isotrivial rational maps defined over the function field mathbb c x of a normal projective variety x in terms of the growth of the degree of the multipliers

the set of maps f_ ɑ b x x ɑ b 2π sin 2πx with any given rotation interval is contractible consider the two parameter family of real analytic maps f b x x b 2 sin 2 x which are lifts of degree one endomorphisms of the circle the purpose of this paper is to provide a proof that for any closed interval i the set of maps f b whose rotation interval is i form a contractible set

on the connectivity of the julia sets of meromorphic functions we prove that every transcendental meromorphic map f with disconnected julia set has a weakly repelling fixed point this implies that the julia set of newton s method for finding zeroes of an entire map is connected moreover extending a result of cowen for holomorphic self maps of the disc we show the existence of absorbing domains for holomorphic self maps of hyperbolic regions whose iterates tend to a boundary point in particular the results imply that periodic baker domains of newton s method for entire maps are simply connected which solves a well known open question

direct numerical simulation of flow past elliptic cylinders flow over elliptic cylinders can be considered prototypical of flow over a range of bluff bodies since the geometry allows one to study the effect of both thickness and angle of attack on the flow field therefore a careful study of this flow should provide valuable insight into the phenomenon of unsteady separation and the structure of bluff body wakes with this in mind a spectral collocation technique has been developed to simulate the full three dimensional incompressible flow over elliptic cylinders and unlike spectral element and spectral multidomain techniques here the flow is solved in a single domain the equations are discretized on a body fitted elliptic cylindrical grid and properties of the metric associated with this coordinate system are used to solve the governing equations in an efficient manner key issues including the inflow and outflow boundary conditions and time discretization are discussed in detail with the hope that this will facilitate future simulations of similar flows finally we present results of two and three dimensional simulations for a range of flow and geometric parameters the results are compared with available experimental data and it is found that important quantities like strouhal numbers and drag coefficients match well with established values

learning from multiple sources we consider the problem of learning accurate models from multiple sources of nearby data given distinct samples from multiple data sources and estimates of the dissimilarities between these sources we provide a general theory of which samples should be used to learn models for each source this theory is applicable in a broad decision theoretic learning framework and yields general results for classification and regression a key component of our approach is the development of approximate triangle inequalities for expected loss which may be of independent interest we discuss the related problem of learning parameters of a distribution from multiple data sources finally we illustrate our theory through a series of synthetic simulations

ergodic and bernoulli properties of analytic maps of complex projective space we examine the measurable ergodic theory of analytic maps f of complex projective space we focus on two different classes of maps ueda maps of p n and rational maps of the sphere with parabolic orbifold and julia set equal to the entire sphere we construct measures which are invariant ergodic weak or strong mixing exact or automorphically bernoulli with respect to these maps we discuss topological pressure and measures of maximal entropy h f h top f log deg f we find analytic maps of p 1 and p 2 which are one sided bernoulli of maximal entropy including examples where the maximal entropy measure lies in the smooth measure class further we prove that for any integer d 1 there exists a rational map of the sphere which is one sided bernoulli of entropy log d with respect to a smooth measure

a pseudospectral matrix element method for solution of three dimensional incompressible flows and its parallel implementation abstract a new pseudospectral matrix element psme method employing the primitive variable formulation of the navier stokes equations was used to simulate 3 dimensional time dependent driven cavity flow at a reynolds number of 3200 with an aspect ratio of 3 in the spanwise direction as well as 3 dimensional flow over a backward step the new method only requires functions which are c 0 continuous across the interface between two adjacent elements it also ensures that the continuity equation is satisfied everywhere in the interior including the inter element points and on the boundary the resulting complex geometry for flow over a backward step can be divided into a number of overlapping subdomains by a domain decomposition of simpler geometry with patched grid points in which the solution is more easily obtained with an iterative procedure between subdomains the complete solution is found by the schwarz alternating procedure sap with an eigenfunction expansion for the pressure storage requirements for the 3d inversion step o n 6 are reduced to o n 3 if the inverse of pressure equation is not stored the parallel implementation of the three most time consuming procedures i computing the partial derivatives of scalar fields in terms of dotproduct ii transforming between eigenfunction space and physical space for the pressure in terms of matrix multiplications and iii performing the forward and backward sweeps of an lu decomposition to solve for the pressure have been efficiently performed on a parallel computer the numerical results have reproduced dynamic longitudinal taylor gortler like tgl vortices in qualitative agreement with the experimental results of koseff et al and also indicated other 3 dimensional effects on the flow development computational results for both 2 and 3 dimensional flow over a backward facing step at different reynolds numbers are also presented in this paper no pronounced 3 dimensional effects are observed for reynolds numbers up to 450 except in the boundary layer along the spanwise direction

finite element solution of the 3d compressible navier stokes equations by a velocity vorticity method this paper presents an alternative formulation to the primitive variable form of the navier stokes equations the approach is based on the use of the velocity and vorticity variables as a logical extension of the stream function vorticity method which is quite effective for two dimensional flows second order equations are obtained for the variables and discretization is through a weak galerkin finite element method followed by newton linearization the velocity and vorticity components are solved simultaneously by a direct solver the scheme is demonstrated for two dimensional and three dimensional incompressible and subsonic internal enclosed flow problems

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

on the approximation of the unsteady navier stokes equations by finite element projection methods this paper provides an analysis of a fractional step projection method to compute incompressible viscous flows by means of finite element approximations the analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method such a theoretical distinction leads to a finite element projection method with a poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm error estimates in finite time are given an extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method

a pseudospectral matrix element method for calculation of double diffusive layers near corners abstract a numerical simulation of the time evolving layered structure in a fluid supporting both temperature and salinity gradients was conducted in an l shaped region where an initially stable salt stratified fluid was disturbed by heating the wall containing the corner protruding into the flow in order to resolve the interfacial gradient between each layer of the flow a recently developed pseudospectral element matrix method using the primitive variable formulation of the navier stokes equations was incorporated into a new implementation of the schwarz alternating procedure for navier stokes solutions which differs from the conventional technique this new procedure directly employs the continuity equation at the edge of overlapped domains as the boundary condition for the pressure rather than a dirichlet condition a novel feature of this method is that it allows the use of a previously developed eigenfunction expansion of the pressure operator to reduce the original complex two dimensional problem to a simple one dimensional problem the computational results show that not only are convective layers formed along the vertical wall above the corner but surprisingly the layer emanating from the corner region dominates the development of the upper and lower layers

distribution of rational maps with a preperiodic critical point let f lambda be a family of rational maps of a fixed degree with a marked critical point c lambda under a natural assumption we first prove that the hypersurfaces of parameters for which c lambda is periodic converge as a sequence of positive closed 1 1 currents to the bifurcation current attached to c and defined by demarco we then turn our attention to the parameter space of polynomials of a fixed degree d by intersecting the d 1 currents attached to each critical point of a polynomial bassaneli and berteloot obtained a positive measure mu rm bif of finite mass which is supported on the connectedness locus they showed that its support is included in the closure of the set of parameters admitting d 1 neutral cycles we show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials i e of misiurewicz points

14 velocity and 18 velocity multiple relaxation time lattice boltzmann models for three dimensional incompressible flows in this paper 14 velocity and 18 velocity multiple relaxation time mrt lattice boltzmann lb models are proposed for three dimensional incompressible flows these two models are constructed based on the incompressible lbgk model proposed by he et al 2004 and the mrt lb model proposed by d humieres et al 2002 which have advantages in the computational efficiency and stability respectively through the chapman enskog analysis the models can recover to three dimensional incompressible navier stokes equations in the low mach number limit to verify the present models the steady poiseuille flow unsteady pulsatile flow and lid driven cavity flow in three dimensions are simulated the simulation results agree well with the analytical solutions or the existing numerical results moreover it is found that the present models show higher accuracy than d humieres et al model and better stability than he et al model

a velocity stream function method for three dimensional incompressible fluid flow abstract we describe a velocity stream function method for computing incompressible fluid flow extending earlier work in two to three dimensions we present a strictly divergence free finite element in primitive variables in 3d as the curl of a generalized hermite vector potential element with velocity degrees of freedom and show a method for its derivation this linear velocity element is defined on a reference cube with 8 nodes each node having 3 vector potential stream function and 3 velocity degrees of freedom the method and element are applied to the lid driven cavity problem and open duct flow in three dimensions

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

a mixed basis spectral projection method a new grid less no collocation spectral projection method is presented the unsteady navier stokes equations are approximated according to the variational framework of guermond and quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half steps of the fractional step method but retains only one in the final solution algorithm two different bases built on legendre polynomials are used for the velocity and pressure to solve the corresponding helmholtz and poisson equations by direct spectral elliptic solvers interpolations pn and pn 2 are employed for velocity and pressure to satisfy the lbb stability requirement and a gauss legendre quadrature formula with 3 2n integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms a bdf second order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem

stability analysis of a parametric family of iterative methods for solving nonlinear models a one parametric family of fourth order iterative methods for solving nonlinear systems is presented proving the fourth order of convergence of all members in this family except one of them whose order is five the methods in our family are numerically compared with other known methods in terms of the classical efficiency index order of convergence and number of functional evaluations and in terms of the operational efficiency index which also takes into account the total number of product quotients per iteration in order to analyze its stability and its dynamical properties the parameter space for quadratic polynomials is shown the stability of the strange fixed points is studied in this case we note that even for this particular case the family presents a very interesting dynamical behavior the analysis of the parameter plane allows us to find values for the involved parameter with good stability properties as well as other values with bad numerical behavior finally amongst the first ones there is a special value of the parameter related to a fifth order method in the family

semi implicit discontinuous galerkin methods for the incompressible navier stokes equations on adaptive staggered cartesian grids in this paper a new high order semi implicit discontinuous galerkin method si dg is presented for the solution of the incompressible navier stokes equations on staggered space time adaptive cartesian grids amr in two and three space dimensions the pressure is written in the form of piecewise polynomials on the main grid which is dynamically adapted within a cell by cell amr framework according to the time dependent main grid different face based spatially staggered dual grids are defined for the piece wise polynomials of the respective velocity components although the resulting adaptive staggered grids are more complex than classical uniform cartesian meshes the numerical scheme can still be written in a rather compact form thanks to the use of a tensor product formulation for the definition of the nodal basis in the d dimensional space d 2 3 all the discrete operators can be efficiently written as a combination of linear one dimensional operators acting in the d space directions separately arbitrary high order of accuracy is achieved in space while a very simple semi implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation the real advantages of the staggered grid arise in the solution of the schur complement associated with the saddle point problem of the discretized incompressible navier stokes equations i e after substituting the discrete momentum equations into the discrete continuity equation this leads to a linear system for only one unknown the scalar pressure indeed the resulting linear pressure system is shown to be symmetric and positive definite in order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor an implicit discretization is also used for the diffusive terms in the momentum equation a particular feature of our staggered dg approach is that the viscous stress tensor is discretized on the dual mesh this corresponds to the use of a lifting operator but on the staggered grid both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix free conjugate gradient method for which fast convergence is observed moreover it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner due to the explicit discretization of the nonlinear convective terms the final algorithm is stable within the classical cfl type time step restriction for explicit dg methods based on the fluid velocity moreover by using a suitable local time stepping technique based on the ader dg framework each element can use its optimal local time step also within the amr context the new space time adaptive staggered dg scheme has been thoroughly verified for polynomial degrees up to n 9 for a large set of non trivial test problems in two and three space dimensions for which analytical numerical or experimental reference solutions exist the high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution to the knowledge of the authors this is the first staggered semi implicit dg scheme for the incompressible navier stokes equations on space time adaptive meshes in two and three space dimensions

attracting cycles in p adic dynamics and height bounds for postcritically finite maps a rational function of degree at least 2 with coefficients in an algebraically closed field is postcritically finite pcf if and only if all of its critical points have finite forward orbit under iteration we show that the collection of pcf rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers once the well understood family known as flexible lattes maps is excluded as a consequence there are only finitely many conjugacy classes of non lattes pcf rational maps of a given degree defined over any given number field the key ingredient of the proof is a nonarchimedean version of fatou s classical result that every attracting cycle of a rational function over c attracts a critical point

on the infinite dimensional qr algorithm spectral computations of infinite dimensional operators are notoriously difficult yet ubiquitous in the sciences indeed despite more than half a century of research it is still unknown which classes of operators allow for the computation of spectra and eigenvectors with convergence rates and error control recent progress in classifying the difficulty of spectral problems into complexity hierarchies has revealed that the most difficult spectral problems are so hard that one needs three limits in the computation and no convergence rates nor error control is possible this begs the question which classes of operators allow for computations with convergence rates and error control in this paper we address this basic question and the algorithm used is an infinite dimensional version of the qr algorithm indeed we generalise the qr algorithm to infinite dimensional operators we prove that not only is the algorithm executable on a finite machine but one can also recover the extremal parts of the spectrum and corresponding eigenvectors with convergence rates and error control this allows for new classification results in the hierarchy of computational problems that existing algorithms have not been able to capture the algorithm and convergence theorems are demonstrated on a wealth of examples with comparisons to standard approaches that are notorious for providing false solutions we also find that in some cases the iqr algorithm performs better than predicted by theory and make conjectures for future study

dynamics of a new family of iterative processes for quadratic polynomials in this work we show the presence of the well known catalan numbers in the study of the convergence and the dynamical behavior of a family of iterative methods for solving nonlinear equations in fact we introduce a family of methods depending on a parameter m n 0 these methods reach the order of convergence m 2 when they are applied to quadratic polynomials with different roots newton s and chebyshev s methods appear as particular choices of the family appear for m 0 and m 1 respectively we make both analytical and graphical studies of these methods which give rise to rational functions defined in the extended complex plane firstly we prove that the coefficients of the aforementioned family of iterative processes can be written in terms of the catalan numbers secondly we make an incursion into its dynamical behavior in fact we show that the rational maps related to these methods can be written in terms of the entries of the catalan triangle next we analyze its general convergence by including some computer plots showing the intricate structure of the universal julia sets associated with the methods

newton s method on the complex exponential function we show that when newton s method is applied to the product of a polynomial and the exponential function in the complex plane the basins of attraction of roots have finite area

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

structural instability of exponential functions we first prove some equivalent statements on stability of families of critically finite entire functions then with these in hand a conjecture con cerning stability of the family of exponential functions is affirmatively answered in some cases

chebyshev expansion methods for the solution of the extended graetz problem abstract the extended graetz problem a classical problem described by an equation with anon self adjoint second order elliptic differential operator is solved by two numerical methods based on chebyshev expansions a chebyshev finite difference method and a chebyshev finite element method the latter relies entirely on chebyshev expansions global in the bounded radial coordinate direction and local in the unbounded axial direction and proves to be more accurate than the chebyshev finite difference method in resolving singularities both methods can be applied to general type boundary value problems involving second order elliptic operators have accuracy comparable to high order finite difference schemes and are operation cost efficient

bifurcation measures and quadratic rational maps we study critical orbits and bifurcations in the curves per1 within the moduli space m2 of quadratic rational maps f p 1 p 1 for each 2 c the curve per1 is dened by the condition that the map has a xed point of multiplier we prove that the curve per1 contains innitely many postcritically n ite maps if and only if 0 addressing a special case of bd2 conjecture 1 4 we also show that the two critical points of f dene distinct bifurcation measures along per 1

prescribed cycles of konig s method for polynomials abstract we consider konig s method for finding roots of polynomials let k f n be the function defined in konig s method for polynomial f of order n n 2 for any given complex number c and any given set z 1 z 2 z k of k k 2 distinct complex numbers we present a procedure of constructing a polynomial f n for which is a k cycle of k f n n with multiplier

three dimensional fluid flow calculations using a flux spline method this paper reports the application of a flux spline method to three dimensi onal fluid flow problems the performance of the method is compared to that of a first order differencing scheme the numerical results are compared with a reference solution available in the literature the flux spline scheme is shown to be significantly more accurate than the first order schemes for a prescribed accuracy the flux spline scheme requires a much smaller number of grid points

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

methods with prefixed order for approximating square roots with global and general convergence a family of newton like methods is constructed to approximate the square root of a positive real number the iterative methods of the family are global or generally convergent depending on the prefixed order of convergence is even or odd we show the dynamical behaviour of some of these methods by means of several julia sets and the intricate fractal structures which arise from the order of the iterative methods are displayed

braiding of the attractor and the failure of iterative algorithms le mouvement de l attracteur dans une famille stable d applications rationnelles determine un groupe de tresses sur la sphere de riemann on montre que le groupe correspondant des classes d application est soit reductible soit fini soit fixe l un des attracteurs ceci conduit a un critere topologique d existence des bifurcations dans les algorithmes iteratifs de recherche de racines bases sur des applications rationnels

stability study of eighth order iterative methods for solving nonlinear equations in this paper we study the stability of the rational function associated to a known family of eighth order iterative schemes on quadratic polynomials the asymptotic behavior of the fixed points corresponding to the rational function is analyzed and the parameter space is shown in which we find choices of the parameter for which there exists convergence to cycles or even chaotical behavior showing the complexity of the family moreover some elements of the family with good stability properties are obtained

on general convergence in extracting radicals via a fundamental family of iteration functions newton s method is well known to be generally convergent for solving x n c 0 in this paper we first extend this result to the next two members of an infinite family of high order methods referred to here as the basic family which starts with newton s method while computing roots of unity numerically is a trivial task studying the general convergence of the basic family in this simple case is an important first step toward the understanding of the global behavior of this fundamental family with the aid of polynomiography techniques for the visualization of polynomial root finding we further conjecture the general convergence of all members of the basic family when extracting radicals using the computer algebra system maple we obtain some partial results toward the proof of our conjecture

three dimensional analysis of flow in the spiral casing of a reaction turbine using a differently weighted petrov galerkin method abstract a weighted residual based finite element method has been used to predict the flow structure and the head loss in the spiral casing of a reaction turbine an explicit eulerian velocity correction scheme has been employed to solve the complete reynolds averaged navier stokes equations a differently weighted petrov galerkin dwpg method has been used for spatial discretization the simulation has been performed for high reynolds numbers the head loss in the spiral casing has been calculated for re 5 105 and re 106 the velocity field and the pressure distribution inside the spiral casing corroborate the trends available in literature the velocity field reveals the existence of a free vortex character of the global flow field a strong secondary flow is evolved on the cross stream plane due to the centrifugal forces which act at right angles to the main flow

are there any julia sets for the laguerre iteration function abstract for polynomials some of whose zeros are complex little is known about the overall convergence properties of the laguerre s method the existence of free critical points of the laguerre function as applied to one parameter families of quadratic and cubic polynomials is examined with the help of microcomputer plots we investigate the julia sets of the laguerre function in the special case wherein this function is constructed to converge to the nth roots of unity

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

three dimensional flow in cavity at yaw abstract this study introduces a building block prototype model of three dimensional 3 d flows about protrusions and cavities with an arbitrary angle between the external flow and rigid elements the considered flow in a 3 d cavity has the lid of cavity moving along the lid diagonal case a the proposed case is taken as a typical representative of essentially 3 d highly separated vortical flows having simple single block rectangular geometry of computational domain this 3 d lid driven cavity flow is studied by numerical solution of the 3 d viscous navier stocks equations using fluent software with second order upwind schemes for convective terms computational results are compared to the classical benchmark case where the lid moves in the direction parallel to the cavity side walls case b the volume and plane integrals of momentum in the direction of lid motion and perpendicular to it are introduced to compare cases in a quantitative way the integral momentum of the secondary flow transverse to the lid motion appears to be about an order of magnitude larger than that in case b in the proposed flow the number and strength of secondary vortices depend substantially on the re number of the flow whereas in case b only one pair of secondary vortices appears for arbitrary re number for a rectangular 3 d 3 1 1 cavity the integral momentum of the secondary flow in case a is an order of magnitude larger than that in the case b the flowfield in case a includes a curvilinear separation line and non symmetrical vortices

algorithmic stability and meta learning a mechnism of transfer learning is analysed where samples drawn from different learning tasks of an environment are used to improve the learners performance on a new task we give a general method to prove generalisation error bounds for such meta algorithms the method can be applied to the bias learning model of j baxter and to derive novel generalisation bounds for meta algorithms searching spaces of uniformly stable algorithms we also present an application to regularized least squares regression

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

shepard s interpolation for solution adaptive methods in numerical simulations of fluid dynamics problems solution adaptive methods have proven to be very powerful the implementation of the modified shepard s interpolation to the structured grids used in cfd is suggested and described in this paper which takes advantage of the logical grid structure this technique which is demonstrated to be robust efficient smoother and a more accurate alternative to linear interpolation is used in the remapping step during the solution adaption procedure applications to the solutions of the incompressible navier stokes equations both 2d and 3d are included

how to find all roots of complex polynomials by newton s method we investigate newton s method to find roots of polynomials of fixed degree d appropriately normalized we construct a finite set of points such that for every root of every such polynomial at least one of these points will converge to this root under newton s map the cardinality of such a set can be as small as 1 11 d log2d if all the roots of the polynomial are real it can be 1 30 d

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

effective vorticity velocity formulations for three dimensional incompressible viscous flows the vorticity velocity formulation for incompressible viscous flows is studied the main concern is effective numerical solution of the kinematic cauchy riemann equations for velocity especially when the divergence free condition of vorticity is violated due to numerical error the mathematical formulation of a differential and an integral approaches are revisited a novel projection of vorticity onto the divergence free space and its application to the two approaches are studied as an illustration of the differential approach the 3d lid driven cavity problem is solved the projection scheme is justified through numerical tests by comparing it to a new fully divergence free scheme numerical tests indicate that keeping vorticity divergence free is important for obtaining correct and accurate solutions the theory and methods apply to other divergence free fields of physical interest as well such as that in electro magnetic dynamics