fourier chebyshev methods for the incompressible navier stokes equations in infinite domains a fully spectral numerical scheme is presented for the unsteady high reynolds number incompressible navier stokes equations in domains which are infinite or semi infinite in one dimension the domain is not mapped and standard fourier or chebyshev expansions can be used the handling of the infinite domain does not introduce any significant overhead the scheme assumes that the vorticity in the flow is essentially concentrated in a finite region which is represented numerically by standard spectral collocation methods to accommodate the slow exponential decay of the velocities at infinity extra expansion functions are introduced which are handled analytically a detailed error analysis is presented and two applications to direct numerical simulation of turbulent flows are discussed in relation with the numerical performance of the scheme

reidemeister torsion surgery invariants and spherical space forms the paper gives methods for calculating the surgery obstruction of normal maps with target a finite pd complex with finite fundamental group the results are applied to the spherical space form problem and give fairly complete answers to the questions of dimensional bounds in terms of equivalent specific statements in algebraic number theory first we give a description of the general results for evaluating surgery obstructions to save space we outline the odd dimensional cases only let m x be a degree 1 normal map in odd dimension n and let n n x by duality x is weakly simple and there is a well defined invariant x f e l n zn it is mapped into the actual surgery invariant l f e l h n zn by the map j in the rothenberg exact sequence

fast adaptive 2d vortex methods we present a new approach to vortex methods for the 2d euler equations we obtain long time high order accuracy at almost optimal cost by using three tools fast adaptive quadrature rules a free lagrangian formulation and a useful new analysis of the consistency error our error analysis halves the order of differentiability required of the flow and suggests an efficient new balance of smoothing parameters which works well with fast summation schemes numerical experiments with our methods confirm our theoretical predictions and display excellent long time accuracy

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

the phase diagram of the o n model we investigate the properties of the zero field o n model on the honeycomb lattice outside the range of presently available exact solutions to this purpose we make use of finite size scaling and transfer matrix techniques our results confirm the expected phase diagram of the o n model in the range 0 n 2 it consists of a disordered phase separated by a second order transition from a low temperature phase with algebraically decaying correlations we confirm that in this low temperature phase the exponents are independent of the temperature more significantly we find evidence for this same universal behaviour even in the case n 2 this contrasts with the temperature dependent exponents of the generic xy model our numerical evidence in this case is supported by arguments based on a mapping from the o n 2 model to the ashkin teller model on the triangular lattice furthermore our results agree with the absence of a phase transition for n 2 for negative values of n we observe unphysical singularities in the free energies of finite systems

exact multicritical behaviour of the potts model a two dimensional q state potts model with vacancies and four spin interactions is studied the parameter space of the model contains a critical and a tricritical manifold moreover for 0 or q or 9 4 a multicritical point is found which is the locus where the tricritical transition changes from first to second order at this multicritical point which can be located exactly the model is solvable the authors compute the value of the central charge and a number of critical exponents

2d vortex methods and singular quadrature rules a new high order vortex method for the 2d euler equations is presented the method eliminates smoothing by constructing a singular quadrature rule for the biot savart law at each time step using quadtrees and orthogonal polynomials theory and numerical experiments show that the method is accurate and efficient yielding excellent long term accuracy in almost optimal cpu time

order parameters of the dilute a models the free energy and local height probabilities of the dilute a models with borken 2 symmetry are calculated analytically using inversion and corner transfer matrix methods these models possess four critical branches the first two branches provide new realizations of the unitary minimal series and the other two branches give a direct product of this series with an ising model we identify the integrable perturbations which move the dilute a models away from the critical limit generalized order parameters are defined and their critical exponents extracted the associated conformal weights are found to occur on the diagonal of the relevant kac table in an appropriate regime the dilute a3 model lies in the universality class of the ising model in a magnetic field in this case we obtain the magnetic exponent 15 directly without the use of scaling relations

domain growth kinetics and symmetry breaking in a thermally quenched system the authors simulate thermal quenches on a two dimensional hard core model with symmetry breaking phase transition by dividing large configurations into many small sub squares lack of self averaging is avoided and statistics are improved dynamic scaling is observed during more than two decades

the optimization of convergence for chebyshev polynomial methods in an unbounded domain by using the method of steepest descent i have compared the suitability of three different methods for solving problems in a semi infinite or infinite domain using chebyshev polynomials exponential mappings are uniformyly bad domain truncation and algebraic mapping both work well but each is superior for a different category of problems when the solution is an entire function then domain truncation is best it is always simpler to apply than algebraic mapping and always at least as accurate much more accurate for functions which decay very rapidly for singular functions on the other hand algebraic mapping is better because it is less sensitive to the mapping scale factor l permitting a better compromise in resolving both the singularity and the exponential decay for both types of problems i give simple explicit estimates of both the optimum choise of domain size or mapping factor and of the attainable accuracy for the model entire function exp azk on a semi infinite interval 0 the optimum domain size is l 0 896 na 1k and the smallest attainable error is roughly e n 0 896 k where n is the number of chebyshev polynomals used similar formulas for singular functions handled via algebraic mapping are given in 3 41 to 3 45 below

a spectral method for polar coordinates a new set of polynomial functions that can be used in spectral expansions of c functions in polar coordinates r is defined by a singular sturm liouville equation with the use of the basis functions the spectral representations remain analytic at the pole despite the coordinate singularity because the pole condition is exactly satisfied at the origin for ag azimuthal modes not just a few of the gravest modes which is the usual case based on recurrence relations fast and stable numerical operators for 1 r r d dr the laplacian and helmholtz operators and their inverses are developed although the spacings in the azimuthal direction of the collocation points near the origin are small i e 1 m2 where m is the number of radial modes the explicit numerical method for euler s equation is not stiff at the origin namely the cfl number is o 1 where the grid size in is defined as m i e the maximum allowable timestep is proportional to 1 m not 1 m2

accuracy analysis of a bessel spectral function method for the solution of scattering equations abstract a bessel function spectral method for obtaining the quantum mechanical scattering wave functions developed previously was tested for the case that the potentials are complex gaussian functions since in this particular case the required overlap integrals can be evaluated with great accuracy the dependence of the elastic scattering s matrix on the size of the space of the bessel basis on the choice of the sub set of the basis functions and on the choice of the matching radius was examined the analysis is carried out to an accuracy of eight significant figures

locus of the tricritical transition in a two dimensional q state potts model a simple two dimensional potts model with vacancies is introduced the phase transition between paramagnetic and ferromagnetic is located by simple symmetry arguments it is shown that the transition has a continuous and a first order segment separated by a tricritical transition the locus of the tricritical point is determined by a transformation of the partition sum into that of an o n model at the tricritical point and at some of the critical points this potts model is soluble

spectral methods for the navier stokes equations with one infinite and two periodic directions abstract two numerical methods were designed to solve the time dependent three dimensional incompressible navier stokes equations in boundary layers method a semi infinite domain and mixing layers or wakes method b fully infinite domain their originality lies in the use of rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions combined with one method a or two method b slowly decaying extra functions for each wave vector that exactly represent the irrotational component of the solution at large distances both methods eliminate the pressure term as part of the formulation thus avoiding fractional step time integration they yield rapid convergence and are free of spurious modes in the orr sommerfeld spectra they are also efficient although the operation count is of order n 2 n is the number of modes in the infinite direction these methods have been used for extensive direct numerical simulations of transition and turbulence a new time integration scheme with low storage requirements and good stability properties is also described

discrete orthogonal function expansions for non uniform grids using the fast fourier transform abstract discrete orthogonal function expansions are obtained for non uniformly spaced data by combined use of mapping functions and the fast fourier transform the result yields alias free differentiation and integration operators the method is applicable to periodic zero or zero derivative boundary conditions or combinations thereof truncation is required to avoid aliasing and or singularity but the truncation error is explicit quantitatively expressible and generally small this approach is demonstrated in an infinite physical domain with application to linear convection and diffusion the resulting errors are orders of magnitude smaller than those generated by standard finite difference methods difficulties arising from image fields are an important phenomena the method of differentiation as well as the boundary conditions imply artificial image fields a vortex pairing problem is presented which shows that image flows can totally alter the solution the new scheme avoids this undesirable effect by keeping image flows infinitely far away

the rate of convergence of fourier coefficients for entire functions of infinite order with application to the weideman cloot sinh mapping for pseudospectral computations on an infinite interval we analytically compute the asymptotic fourier coefficients for several classes of functions to answer two questions the numerical question is to explain the success of the weideman cloot algorithm for solving differential equations on an infinite interval their method combines fourier expansion with a change of coordinate using the hyperbolic sine function the sinh mapping transforms a simple function like exp z2 into an entire function of infinite order this raises the second analytical question what is the fourier rate of convergence for entire functions of an infinite order the answer is sometimes even slower than a geometric series in this case the fourier series converge only on the real axis even when the function u z being expanded is free of singularities except at infinity earlier analysis ignored stationary point contributions to the asymptotic fourier coefficients when u z had singularities off the real z axis but we show that sometimes these stationary point terms are more important than residues at the poles of u z

steady state solutions of the euler equations in two dimensions rotating and translating v states with limiting cases i numerical algorithms and results new second and third order algorithms are presented for calculating translating and rotating steady state solutions of the 2d incompressible euler equations which we call v states these are piecewise constant regions of vorticity and the contours bounding them are obtained by solving iteratively a nonlinear integro differential equation new limiting contours with corners are obtained and compared with local analytical solutions the precise results correct mistakes for limiting contours that were previously given

local surgery obstructions and space forms the space form problem concerning the existence of free actions of finite groups on spheres has now been substantially solved except in dimension three i 3 4 8 11 13 14 this paper gives an effective method for the uniqueness question which homotopy types are realized by such actions our results are fairly complete for 2 hyperelementary periodic groups and give new necessary conditions for actions even on s 3 in particular corollary c below eliminates certain homotopically non linear actions which presented an obstacle to knowing precise dimensional bounds see 4 conjecture d the methods also apply to the euclidean space form problem of semi free actions on i k fixing i k 2 where the linear model is a free representation direct sum with a trivial representation a free action on s 1 yields a semi free topological action on l by coning the method involves the calculation of a certain local surgery obstructions w167 2 3 the basic cases are the 2 hyperelementary type i groups extensions of the form l z m z 2k l where m is odd we write o for a sylow 2 subgroup of z and identify it with z 2k let t be the twisting defining the extension and let t cr im t z 2 k aut z m 7 2 z

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

a spectral method for unbounded domains a spectral method for an unbounded domain is presented rational basis functions which are algebraically mapped legendre functions are used for expansion in the radial direction of polar coordinates r or r z they satisfy the pole condition exactly at the coordinate singularity and their behavior asr is suitable for expanding smooth functions which decay algebraically or exponentially asr the method is not stiff when it is applied to initial value problems despite the presence of the coordinate singularity solenoidal vector fields are treated efficiently by the toroidal and poloidal decomposition which reduces the number of dependent variables from 3 to 2 examples include the computation of vortex dynamics in two and three dimensions