construction of constant mean curvature n noids from holomorphic potentials using a weierstrass type representation of constant mean curvature surfaces we give a general method for constructing constant mean curvature n noids of genus 0 from holomorphic potentials where n 3 the ends of these surfaces are embedded and asymptotically approach delaunay surfaces while the surfaces are in general not even almost embedded in particular a 3 parameter family of constant mean curvature trinoids is constructed

higher order approximation methods for the boltzmann equation a higher order time differencing method for the spatially nonhomogeneous boltzmann equation is derived from the integral form of the equation along its characteristic line similar to the splitting method which solves the collisionless equation in the convection step and the spatially homogeneous boltzmann equation in the collision step the present method consists of two steps one of which is the same as the convection step in the splitting method the difference from the splitting method is in the other step where not only the collision term but also its variation along the characteristic line is taken into account correctly the truncation error of this method per time step tiso t3 and its higher order accuracy is demonstrated numerically in the shock propagation problem using the bgk model equation it is shown that such accuracy is never realized in the framework of the conventional splitting formulation which is contrary to bogomolov s result u s s r comput math math phys 28 79 1988 the other higher order methods based on the integral form are also presented furthermore the extension to the stochastic approach modification of the conventional direct simulation monte carlo procedure is proposed

real forms of complex surfaces of constant mean curvature it is known that complex constant mean curvature cmc for short immersions in c 3 are natural complexifications of cmc immersions in r 3 in this paper conversely we consider real form surfaces of a complex cmc immersion which are defined from real forms of the twisted sl 2 c loop algebra asl 2 and classify all such surfaces according to the classification of real forms of asl 2 there are seven classes of surfaces which are called integrable surfaces and all integrable surfaces will be characterized by the lorentz harmonicities of their gauss maps into the symmetric spaces s 2 h 2 s 1 1 or the 4 symmetric space sl 2 u 1 we also give a unification to all integrable surfaces via the generalized weierstrass type representation

actions of loop groups on harmonic maps we describe a general framework in which subgroups of the loop group agin act on the space of harmonic maps from s2 to gln this represents a simplification of the action considered by zakharov mikhailov shabat zm zs in that we take the contour for the riemann hilbert problem to be a union of circles however it reduces the basic ingredient to the well known birkhoff decomposition of agin s and this facilitates a rigorous treatment we give various concrete examples of the action and use these to investigate a suggestion of uhlenbeck uh that a limiting version of such an action completion gives rise to her fundamental process of adding a uniton it turns out that this does not occur because completion preserves the energy of harmonic maps however in the special case of harmonic maps from s2 to complex projective space we describe a modification of this completion procedure which does indeed reproduce adding a uniton one aspect of the theory of symmetry groups for diffierential equations is the idea of proliferation of solutions by this is meant that starting with an obvious solution application of symsmetries can lead to further perhaps less obvious solutions in this paper we shall apply the general framework described by zakharov et al zm zs to the harmonic map equation the principal chiral or sigma model of mathematical physics to put this into a wider context we recall that a fundamental discovery in the study of the korteweg de vries and related equations was that the space of solutions admits a certain infinite dimensional symmetry group historically the first manifestation of this via noether s theorem was the appearance of infinitely many conservation laws later this led to an elegant algebraic description of solutions of the equation as the points of an orbit of a representation of an infinite dimensional group see for example wi for a brief description of some of these ideas the existence of previously unnoticed symmetries hidden symmetries for other equations became an interesting possibility and attempts have been made so far with much less success than in the case of the kdv equation to unearth them one such example is the harmonic map equation for maps from a surface d into a compact lie group g or homogeneous space g h the basic observation po uh zm zs which introduces an infinite dimensional group is that harmonic maps from d to g correspond to certain received by the editors february 9 1990 1980 mathematics subject classification 1985 revision primary 58e20 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

lawson s genus two surface and meromorphic connections we investigate the lawson genus 2 surface by methods from integrable system theory to this surface we derive a globally defined meromorphic dpw potential which is determined up to two unknown functions depending only on the spectral parameter

lattice boltzmann method and gas kinetic bgk scheme in the low mach number viscous flow simulations both lattice boltzmann method lbm and the gas kinetic bgk scheme are based on the numerical discretization of the boltzmann equation with collisional models such as the bhatnagar gross krook bgk model lbm tracks limited number of particles and the viscous flow behavior emerges automatically from the intrinsic particle stream and collisions process on the other hand the gas kinetic bgk scheme is a finite volume scheme where the time dependent gas distribution function with continuous particle velocity space is constructed and used in the evaluation of the numerical fluxes across cell interfaces currently lbm is mainly used for low mach number nearly incompressible flow simulation for the gas kinetic scheme the application is focusing on the high speed compressible flows in this paper we are going to compare both schemes in the isothermal low mach number flow simulations the methodology for developing both schemes will be clarified through the introduction of operator splitting boltzmann model and operator averaging boltzmann model from the operator splitting boltzmann model the error rooted in many kinetic schemes which are based on the decoupling of particle transport and collision can be easily understood as to the test case we choose to use the 2d cavity flow since it is one of the most extensively studied cases detailed simulation results with different reynolds numbers as well as the benchmark solutions are presented

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

numerical hydrodynamics from gas kinetic theory a new high resolution numerical hydrodynamic scheme is developed from considerations of gas kinetic theory this boltzmann type scheme uses the particle distribution function and follows its evolution to evaluate the numerical fluxes the intrinsic multidimensional property in the evolution stage of gas distribution function distinguishes it from any other high order upwind scheme from approximate boltzmann equation the navier stokes equations can be derived and this new numerical method uses this advantage to approximate the solutions of the navier stokes equations where the advection and dissipation effects are coupled in the time dependent gas distribution function several well defined numerical test cases are presented in one and two dimensions and it is shown that the resolution in the numerical results is competitive with those of any of the current high resolution schemes without special treatment of the discontinuities for strong rarefaction waves our scheme gives better results in this thesis we try to separate clearly numerical and physical effects and we also give a concise explanation of the necessity of including numerical viscosities implicitly or explicitly in high resolution godunov type and central difference schemes even for the navier stokes equations the main part of this thesis will be published in two papers 27 and 45 in the journal of computational physics

a molecular collision operator of adjustable direction for the discrete velocity direction model abstract the discrete velocity direction model is an approximate method to the boltzmann equation a developed molecular collision operator for the model is presented in this paper under the new operator the discrete directions of molecules are adjustable namely both the number and the angles of discrete directions can be changed as needed in the discrete velocity direction model at the same time the governing equations will keep unchanged when the number of discrete directions changes in fact with the continuous molecular speed the discrete velocity direction model has been able to employ any discrete velocities in numerical calculations the discrete velocity direction model under the new collision operator was applied into some benchmark flows in micro scales in this paper and the influence of the number of discrete velocities on the computational accuracy was analyzed the numerical results show that the accuracy of the discrete velocity direction model can be improved significantly by employing more discrete directions especially for the gas flows at large knudsen number with appropriate discrete velocities this model has been able to give accurate numerical results in all flow regimes in addition it is proved that the discrete velocity direction model under the new collision operator satisfies a global h theorem unconditionally which means that the new operator further improves the intrinsic stability of the discrete velocity direction model

simulation of cavity flow by the lattice boltzmann method a detailed analysis is presented to demonstrate the capabilities of the lattice boltzmann method thorough comparisons with other numerical solutions for the two dimensional driven cavity flow show that the lattice boltzmann method gives accurate results over a wide range of reynolds numbers studies of errors and convergence rates are carried out compressibility effects are quantified for different maximum velocities and parameter ranges are found for stable simulations the paper s objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers

cyclicity of elliptic curves modulo p and elliptic curve analogues of linnik s problem let e be an elliptic curve defined over and of conductor n for a prime we denote by the reduction of e modulo p we obtain an asymptotic formula for the number of primes p x for which is cyclic assuming a certain generalized riemann hypothesis the error terms that we get are substantial improvements of earlier work of j p serre and m ram murty we also consider the problem of finding the size of the smallest prime p p e for which the group is cyclic and we show that under the generalized riemann hypothesis p e log n 4 if e is without complex multiplication and p e log n 2 if e is with complex multiplication for any 0 1

generalized dpw method and an application to isometric immersions of space forms let g be a complex lie group and g denote the group of maps from the unit circle mathbb s 1 into g of a suitable class a differentiable map f from a manifold m into g is said to be of connection order a b if the fourier expansion in the loop parameter of the mathbb s 1 family of maurer cartan forms for f namely f lambda 1 rm d f lambda is of the form sum i a b alpha i lambda i most integrable systems in geometry are associated to such a map roughly speaking the dpw method used a birkhoff type splitting to reduce a harmonic map into a symmetric space which can be represented by a certain order 1 1 map into a pair of simpler maps of order 1 1 and 1 1 respectively conversely one could construct such a harmonic map from any pair of 1 1 and 1 1 maps this allowed a weierstrass type description of harmonic maps into symmetric spaces we extend this method to show that for a large class of loop groups a connection order a b map for a 0 b splits uniquely into a pair of a 1 and 1 b maps as an application we show that constant non zero sectional curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds reducing the problem at least locally to the flat case to extend the dpw method sufficiently to handle this problem requires a more general iwasawa type splitting of the loop group which we prove always holds at least locally

reductions of an elliptic curve and their tate shafarevich groups in this paper we study the tate shafarevich groups iii p of the reductions modulo primes p of an elliptic curve e considered as being defined over their function fields assuming grh when e has no cm we show that iii p is trivial for a positive proportion of primes p provided e has an irrational point of order two

frobenius fields for elliptic curves let e bbb q be an elliptic curve over the field of rational numbers with mathop rm end bar bbb q e bbb z let k be a fixed imaginary quadratic field over bbb q and x a positive real number for each prime p of good reduction for e let pi p e be a root of the characteristic polynomial of the frobenius endomorphism of e over the finite field bbb f p let pi e k x be the number of primes p leq x such that the field extension bbb q pi p e is the fixed imaginary quadratic field k we present upper bounds for pi e k x obtained using two different approaches the first one inspired from work of serre is to consider the image of frobenius in a mixed galois representation associated to k and to the elliptic curve e the second one inspired from work of cojocaru fouvry and murty is based on an application of the square sieve the bounds obtained using the first approach are better pi e k x ll x 4 5 log x 1 5 and are the best known so far the bounds obtained using the second approach are weaker but are independent of the number field k a property which is essential for other applications all results are conditional upon grh

gas kinetic scheme with discontinuous derivative for low speed flow computation the present paper concerns the improvement of the gas kinetic scheme gks for low speed flow computation in the modified gks scheme the flow distributions with discontinuous derivatives are used as an initial condition at the cell interface for the flux evaluation this discontinuity is determined by considering both the flow characteristic and grid s resolution compared with gks method with a continuous slope for the flow variables at a cell interface the new scheme is more robust and accurate in the under resolved flow computation the new scheme presents much less numerical oscillation the extension of the current scheme to unstructured mesh is straightforward to validate the method both computations of 2d lid driven cavity flow and 3d flow past a sphere are performed the numerical results validate the current method

anomalous diffusion limit induced on a kinetic equation using a compactness argument based on the velocity averaging lemma of golse et al it is shown that the limiting behavior of a kinetic linearized bgk gas model confined between two plates with maxwell boundary conditions when the distance between the plates goes to zero under a suitable anomalous scaling is diffusive we do not require the use of central limit theorems as in the method of borgers et al

finite difference lattice boltzmann method applied to acoustic scattering problems this paper reports on an attempt to simulate acoustic waves scattering using a finite difference lattice boltzmann method based on an alternative lattice equilibrium particle distribution function constructed for compressible thermal fluids the studies focus on acoustics scattering by a zero circulation vortex and by an isolated thermal source with no heat gain loss two limiting cases of each type of scattering are examined one is the case of an incoming acoustic wave with a short wavelength and the other has a relatively long wavelength compared with the characteristic dimension of the obstacle these scattering problems have been treated previously using a conventional lattice boltzmann method and a gas kinetic scheme the results showed that these methods were only able to simulate the short wavelength limit case with fair accuracy for the two types of acoustics scattering considered because the present approach is able to recover the compressible navier stokes equations with correct fluid properties the finite difference solution of the proposed alternative modeled lattice boltzmann equation allows the limiting cases of the acoustics scattering problems to be calculated without numerical instability the results thus obtained are in agreement either with analysis or with results obtained from direct aeroacoustics simulations employing the compressible navier stokes equations

a unified gas kinetic scheme for continuum and rarefied flows iv fluid dynamic equations are valid in their respective modeling scales such as the particle mean free path scale of the boltzmann equation and the hydrodynamic scale of the navier stokes ns equations with a variation of the modeling scales theoretically there should have a continuous spectrum of fluid dynamic equations even though the boltzmann equation is claimed to be valid in all scales many boltzmann solvers including direct simulation monte carlo method require the cell resolution to the order of particle mean free path scale therefore they are still single scale methods in order to study multiscale flow evolution efficiently the dynamics in the computational fluid has to be changed with the scales a direct modeling of flow physics with a changeable scale may become an appropriate approach the unified gas kinetic scheme ugks is a direct modeling method in the mesh size scale and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path the cell size of ugks is not limited by the particle mean free path with the variation of the ratio between the numerical cell size and local particle mean free path the ugks recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale the previous ugks is mostly constructed from the evolution solution of kinetic model equations even though the ugks is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time it still has space to develop more accurate flow solver in the region where the time step is comparable with the local particle mean free time in such a scale there is dynamic difference from the full boltzmann collision term and the model equations this work is about the further development of the ugks with the implementation of the full boltzmann collision term in the region where it is needed the central ingredient of the ugks is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled the newly developed ugks has the asymptotic preserving ap property of recovering the ns solutions in the continuum flow regime and the full boltzmann solution in the rarefied regime in the mostly unexplored transition regime the ugks itself provides a valuable tool for the non equilibrium flow study the mathematical properties of the scheme such as stability accuracy and the asymptotic preserving will be analyzed in this paper as well

boltzmann equation description of flows at long mean free paths the flow of a gas around a rectangular object is simulated by solving the boltzmann equation for the gas the boltzmann equation is solved by means of a method of characteristics which we refer to as a convective scheme cs this paper focuses first on two computational issues we describe how the cs which is presented here is implemented so as to handle reflecting boundary conditions very accurately next a collision operator for the self collisions of the neutral gas has been developed which conserves momentum and energy exactly and which also preserves higher moments of the distribution so as to correctly calculate quantities such as viscosity finally the method is illustrated briefly by calculating flow patterns and drag coefficients for a low mach number flow around a rectangular obstacle over a range of knudsen numbers which spans the transitional regime and very accurate values of the drag coefficient are obtained across the whole range

simulation of flows around an impulsively started circular cylinder by taylor series expansion and least squares based lattice boltzmann method the two dimensional incompressible viscous flow past an impulsively started circular cylinder for a wide range of reynolds numbers re 20 9500 is studied computationally by using an explicit taylor series expansion and least squares based lattice boltzmann method the final equation for distribution function in our method is in an explicit form and essentially has no limitation on choice of mesh structure and lattice model it can be easily applied to simulation of flows with curved boundaries such as the problem considered in this work for the flow past an impulsively started circular cylinder numerical results obtained by present method agree very well with experimental data and computational results of navier stokes equations available in the literature

coarse classification of constant mean curvature cylinders we give a coarse classification of constant mean curvature cmc immersions of cylinders into r 3 via the loop group method particularly for this purpose we consider double loop groups and a new type of potentials which are meromorphic 1 forms on riemann surfaces

hamiltonians and zero curvature equations for integrable partial differential equations we discuss the relationship between two approaches to integrable partial differential equations one using formal affine lie algebras and the other banach lie groups in the first approach integrability of the equations follows from commutativity of hamiltonian flows on the lie algebra while in the second it follows from commutativity of certain flows induced by an action on the banach lie group we show that these two methods are essentially equivalent since one can calculate one type of the flows from the other the classes of solutions encompassed by the two methods however vary significantly we demonstrate this relationship specifically with the nonlinear schrodinger equation

low speed flow simulation by the gas kinetic scheme this paper extends the gas kinetic bgk type scheme to low mach number flows and thus shows that incompressible flow solutions are accurately obtained from the bgk scheme in the low mach number limit the influence of boundary conditions internal molecular degrees of freedomk and the flow mach numbermon the accuracy of the solutions of incompressible or nearly incompressible flow problems is quantitatively evaluated the gas kinetic scheme is tested carefully in two numerical examples namely the cavity flow problem and the flow passing a backward facing step problem for the cavity flow problem the numerical results from the gas kinetic scheme under different reynolds numbers compare well with ghia s data for the backward step problem the numerical results are compared accurately with previously published experimental data

a discrete velocity direction model for the boltzmann equation and applications to micro gas flows a discrete velocity direction model for the boltzmann equation is proposed in this paper which provides an alternative technique to the rarefied gas flows in this model the directions of molecular velocities are discrete which are restricted in eight fixed directions while the molecular speed rate is still continuous by this approximation the boltzmann equation in the six dimensional phase space is replaced by eight differential integral equations in three dimensional space thus the computational cost is reduced greatly by reduction of three dimensions the number of discrete velocities is not fixed in the present model because the speed rate can be truncated arbitrarily this is distinguished from the conventional discrete velocity models dvm to test this technique it was applied to the couette flow and poiseuille flow the computed results agree well with those by the linearized boltzmann equation and the dsmc method

rkdg finite element method combined with bgk scheme for solving fluid dynamics system this paper presents a new type of second order scheme for solving the system of euler equations which combines the runge kutta discontinuous galerkin rkdg finite element method and the bhatnagar gross krook bgk scheme we first discretize the euler equations in space with the dg method and then the resulting system will be discretized using an rk method finally a second order bgk method is used to construct the numerical flux the proposed scheme keeps the main advantages of the dg finite element method including its flexibility in handling irregular solution domains and in parallelization moreover with the bgk approach the proposed schemes avoid using riemann solvers the efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one and two dimensional spaces

gas kinetic finite volume methods flux vector splitting and artificial diffusion abstract in order to improve our understanding of boltzmann type schemes which have been recently proposed for the euler and navier stokes equations by prendergast and xu we have modified the gas kinetic approach and applied it to simple hyperbolic conservation laws we found that the gas kinetic discretization can be explained in terms of standard central difference and upwind schemes artificial viscosity concepts are reviewed and linked to the grid size and the physical length scale of the discontinuity also a new three dimensional gas kinetic scheme for the numerical navier stokes equations whose solution satisfies the entropy condition is presented two numerical limits of the scheme are obtained the first one is the one step lax wendroff scheme and the second one is the kinetic flux vector splitting scheme a new relaxation scheme for steady state calculations is also formulated and implemented in the multigrid time stepping technique of jameson when applied to the euler equations the resulting method yields high accuracy and fast convergence to a steady state