spiralling self avoiding walks an exact solution an exact solution is presented to a problem of spiralling self avoiding walks on the square lattice recently proposed by privman 1983 for n to infinity the number of n step spiral walks increases as cn approximately 2 23 5 4 pi n 7 4 exp 2 pi n 3 1 2 and their root mean square end to end distance behaves as rn approximately 1 2 square root pi 1n1 2 log n

renormalisation theory of the self avoiding levy flight the self avoiding levy flight salf in d dimensions with levy exponent mu is formulated as a geometrical equilibrium statistical mechanical problem a direct renormalisation theory based on modern field theoretic techniques is used to derive the critical exponents and the end to end distance probability function through first order in epsilon 2 mu d the non perturbative structure of the probability function is characterised by a universal scaling function the salf represents a simple many body system that can assume a continuum of values of epsilon near zero

critical behaviour of directed self avoiding walks a new directed self avoiding walk directed saw is proposed and studied on a square lattice the critical behaviour of such directed saws in two dimensions is established and shown to be of different universality class from that of an ordinary saw

helicity and the calugareanu invariant the helicity of a localized solenoidal vector field i e the integrated scalar product of the field and its vector potential is known to be a conserved quantity under frozen field distortion of the ambient medium in this paper we present a number of results concerning the helicity of linked and knotted flux tubes particularly as regards the topological interpretation of helicity in terms of the gauss linking number and its limiting form the c lug reanu invariant the helicity of a single knotted flux tube is shown to be intimately related to the c lug reanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given helicity is decomposed into writhe and twist contributions the writhe contribution involving the gauss integral for definition see equation 4 8 which admits interpretation in terms of the sum of signed crossings of the knot averaged over all projections part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as intrinsic twist of the field lines in the flux tube around the knot see equations 5 13 and 5 15 the generic behaviour associated with the deformation of the knot through a configuration with points of inflexion points at which the curvature vanishes is analysed and the role of the twist parameter is discussed the derivation of the c lug reanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems

critical exponents of directed self avoiding walks it is argued that directed self avoiding random walks are in the same universality class as directed unrestricted random walks and therefore should exhibit mean field exponents gamma 1 nu sub 1 nu perpendicular to 1 2 in all dimensionalities

critical properties of directed self avoiding walks the generating functions and mean displacements of various two dimensional directed self avoiding walk models are calculated exactly by a simple transfer matrix approach asymptotically the authors find r sub n approximately rn2 1 2 approximately n and rperpendicular to n2 1 approximately n1 2 where n is the number of steps in the walk and and perpendicular to refer to projections of the displacement parallel and perpendicular to the preferred axis of the walk respectively some general properties of directed self avoiding walks for arbitrary dimensions are discussed as well

isoperimetric problems for the helicity of vector fields and the biot savart and curl operators the helicity of a smooth vector field defined on a domain in three space is the standard measure of the extent to which the field lines wrap and coil around one another it plays important roles in fluid mechanics magnetohydrodynamics and plasma physics the isoperimetric problem in this setting is to maximize helicity among all divergence free vector fields of given energy defined on and tangent to the boundary of all domains of given volume in three space the biot savart operator starts with a divergence free vector field defined on and tangent to the boundary of a domain in three space regards it as a distribution of electric current and computes its magnetic field restricting the magnetic field to the given domain we modify it by subtracting a gradient vector field so as to keep it divergence free while making it tangent to the boundary of the domain the resulting operator when extended to the l2 completion of this family of vector fields is compact and self adjoint and thus has a largest

spiral growing self avoiding walk a new universality class the author introduces the concept of a spiral growing self avoiding walk sgsaw by imposing privman s spiralling constraint ibid vol 16 p l571 1983 on the growing self avoiding walk introduced by majid et al phys rev lett vol 52 p 1257 1984 and lyklema and kremer ibid vol 17 p l691 1984 he studies the critical properties of the sgsaw using monte carlo simulation the exponent nu associated with the average radius of gyration sn is 0 8 on the other hand the critical exponent nu r associated with the end to end distance rn is much less than nu the latter is in sharp contrast to the result nu nu r for the usual saw directed gsaws are shown to behave like directed usual saws

electrodynamics and the gauss linking integral on the 3 sphere and in hyperbolic 3 space in this first of two papers we develop a steady state version of classical electrodynamics on the 3 sphere and in hyperbolic 3 space including an explicit formula for the vector valued green s operator an explicit formula of biot savart type for the magnetic field and a corresponding ampere s law contained in maxwell s equations we then use this to obtain explicit integral formulas for the linking number of two disjoint closed curves in these spaces these formulas like their prototypes in euclidean 3 space are geometric rather than just topological because their integrands are invariant under orientation preserving isometries of the ambient space in the second paper we obtain integral formulas for twisting writhing and helicity and prove the theorem link twist writhe in the 3 sphere and in hyperbolic 3 space we then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces an ann

an exact formula for the number of spiral self avoiding walks an exact closed form expression is obtained for the number sn of spiral self avoiding walks with n steps on the square lattice this result is used to derive a complete asymptotic expansion for sn which is valid as n to infinity

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

asymptotic properties of spiral self avoiding walks the authors consider the spiral self avoiding walk model recently introduced by privman 1983 on the basis of an analogy with the partitioning of the integers they argue that the number of n step spiral walks should increase asymptotically as rho square root n with rho a constant leading to an essential singularity in the generating function enumeration data to 65 terms indicates however that cn apparently varies as rho n alpha with alpha approximately 0 55 they also study the n dependence of the mean square end to end distance rn2 and of the mean rotation angle theta n for n step walks from series extrapolations they estimate that rn2 approximately n1 2 and theta n approximately n0 55

linking twisting writing and helicity on the 2 sphere and in hyperbolic 3 space in the first paper of this series electrodynamics and the gauss linking integral on the 3 sphere and in hyperbolic 3 space we developed a steady state version of classical electrodynamics in these two spaces including explicit formulas for the vector valued green s operator explicit formulas of biot savart type for the magnetic field and a corresponding ampere s law contained in maxwell s equations and then used these to obtain explicit inte gral formulas for the linking number of two disjoint closed curves in this second paper we obtain integral formulas for twisting writhing and helicity and prove that link twist writhe on the 3 sphere and in hyperbolic 3 space we then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces an announcement of these results and a hint of their proofs can be found in the math arxiv math gt 0406276 while an expanded version of the first paper with full proofs can be found at math gt 0510388

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

the asymptotic form for the number of spiral self avoiding walks the author investigates the asymptotic behaviour of the number sn of spiral self avoiding walks on the square lattice and show that sn approximately rho square root n with exp pi 312 or rho or exp 2 pi 312

the nontriviality of the one dimensional problem of a true self avoiding walk it is shown that the root mean square distance r2 12 for the self avoiding walk problem is related to the number of steps n as r2 approximately n2 nu with nu 2 3 rather than the self obvious nu 1 this coincides with the mean field theory result obtained by pietronero 1983 which erroneously was not extended to the case d 1

renormalisation of the true self avoiding walk it is shown that the renormalisation of the true self avoiding walk of amit et al 1982 is its upper critical dimensionality two involves at least two possibly three coupling constants fixed points to first order in epsilon 2 d are identified and the asymptotic behaviour at d 2 is discussed

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

a special class in vanishing helicity ideal flows the derivation of a lagrangian invariant so called spirality is reviewed through the lagrangian coordinates the value of the spirality is fixed up to a gauge transformation the helicity conservation follows directly from this invariant among all ideal flows with zero helicity in a domain of flow frozen into the fluid motion a special class is introduced this special topological class has the possibility to transform its spirality to be identically zero everywhere in the domain for those simply connected domains of motion a necessary and a sufficient condition is presented for these zero spirality flows

growing partially directed self avoiding walks a partially directed self avoiding walk model with the kinetic growth weighting is solved exactly on the square lattice and for two restricted strip geometries some finite size effects are examined

spiral self avoiding walks self avoiding walks with a spiral constraint on the square lattice are enumerated up to 40 steps numerical evidence suggests that they belong to a new universality class

finite difference methods for 3d viscous incompressible flows in the vorticity vector potential formulation on nonstaggered grids simple efficient and accurate finite difference methods are introduced for 3d unsteady viscous incompressible flows in the vorticity vector potential formulation on nonstaggered grids two different types of methods are discussed they differ in the implementation of the normal component of the vorticity boundary condition and consequently the enforcement of the divergence free condition for vorticity both second order and fourth order accurate schemes are developed a detailed accuracy test is performed revealing the structure of the error and the effect of how the convective terms are discretized near the boundary the influence of the divergence free condition for vorticity to the overall accuracy is studied results on the cubic driven cavity flow at reynolds number 500 and 3200 are shown and compared with that of the mac scheme

trkalian fields and radon transformation we write the spherical curl transformation for trkalian fields using differential forms then we consider radon transform of these fields the radon transform of a trkalian field satisfies a corresponding eigenvalue equation on a sphere in transform space the field can be reconstructed using knowledge of the radon transform on a canonical hemisphere we consider relation of the radon transformation with biot savart integral operator and discuss its transform introducing radon biot savart operator the radon transform of a trkalian field is an eigenvector of this operator we also present an ampere law type relation for these fields we apply these to lundquist solution we present a chandrasekhar kendall type solution of the corresponding equation in the transform space lastly we focus on the euclidean topologically massive abelian gauge theory the radon transform of an anti self dual field is related by antipodal map on this sphere to the transform of the self dual field obtained by inverting space c

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

the growing self avoiding walk the authors introduce a new self avoiding walk with one step probabilities which depend on the local environment as a consequence this walk is irreversible and models the growth process of a linear polymer in a good solvent to calculate its properties they have performed exact enumerations up to 22 steps on the square lattice and on the diamond lattice this gives for the critical indices the values v 0 68 gamma 1 16 in two dimensions and v 0 525 and gamma 1 in three dimensions

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

on the number of spiral self avoiding walks the authors consider the problem of spiral self avoiding walks as recently introduced by privman 1983 they prove that the number of n step spiral self avoiding walks is given by sn exp 2 pi n 3 12 n74 c 1 o 1 square root n where c pi 4 354 similar results for various subsets of these walks are also obtained

diffusion on random systems above below and at their percolation threshold in two and three dimensions a detailed monte carlo study is presented for classical diffusion random walks on random l l triangular and l l l simple cubic lattices with l up to 4096 and 256 respectively the speed of a cyber 205 vector computer is found to be about one order of magnitude larger than that of a usual cdc cyber 76 computer to reach the asymptotic scaling regime walks with up to 10 million steps were simulated with about 10 jl steps in total for l 256 at the percolation threshold we review and extend the dynamical scaling description for the distance traveled as function of time the diffusivity above the threshold and the cluster radius below earlier discrepancies between scaling theory and computer experiment are shown to be due to insufficient monte carlo data the conductivity exponent is found to be 2 0 0 2 in three and 1 28 0 02 in two dimensions our data in three dimensions follow well the finite size scaling theory below the threshold the approach of the distance traveled to its asymptotic value is consistent with theoretical speculations and an exponent 2 5 independent of dimensionality the correction to scaling exponent at pc seems to be larger in two than in three dimensions

on three dimensional self avoiding walks long self avoiding walks of up to 2400 steps have been generated on the sc and bcc lattices using an improved monte carlo technique analysis of the asymptotic length dependence of the end to end distance and radius of gyration of the walks leads to values in the range 0 591 0 593 for the critical exponent nu the simple power law dependence on length provides an excellent fit to the data over walk lengths between 120 and 2400 with fluctuations around the asymptotic results averaging to a mere 0 13 systematic deviations from asymptotic behaviour in short self avoiding walks have also been examined using monte carlo generated walks ranging in length from 12 to 60 the results do not support the existence of the non analytic correction predicted by the renormalisation group in the light of this unexpected result available series expansions for walks on the fcc lattice have been re examined and previous claims to have observed non analytic behaviour questioned equal if not better convergence of the extrapolated series can be obtained without resorting to non analytic correction terms finally an analysis has been made of the radius of gyration series to which several new terms have been added

two dimensional polymers universality and correction to scaling improved monte carlo methods have been used to study the asymptotic behaviour of self avoiding walks on two dimensional lattices the mean square end to end distance rn2 and radius of gyration sn2 are both found to scale as n2v with exponent values very close to the expected v 3 4 and an amplitude ratio that is universal small n deviations of rn2 from the asymptotic results indicate a correction to scaling exponent delta 1 a value that differs from previous estimates

the asymptotic behaviour of directed self avoiding walks chakrabarti and manna 1983 have made a numerical study of directed self avoiding walks on a square lattice in which the random walker is forbidden to move upward extrapolating from an enumeration of n step walks up to n 14 they claimed that the mean end to end displacement rn behaves asymptotically for large n as nnu with nu 0 86 or 0 02 it is shown that a simple application of the generating function method yields the behaviour of n step walks of this type exactly for all n and the result nu 1 is rigorously proved

an explicit divergence free dg method for incompressible flow abstract we present an explicit divergence free dg method for incompressible flow based on velocity formulation only an h div conforming and globally divergence free finite element space is used for the velocity field and the pressure field is eliminated from the equations by design the resulting ode system can be discretized using any explicit time stepping methods we use the third order strong stability preserving runge kutta method in our numerical experiments our spatial discretization produces the identical velocity field as the divergence conforming dg method of cockburn et al 2007 based on a velocity pressure formulation when the same dg operators are used for the convective and viscous parts due to the global nature of the divergence free constraint and its interplay with the boundary conditions it is very hard to construct local bases for our finite element space here we present a key result on the efficient implementation of the scheme by identifying the equivalence of the mass matrix inversion of the globally divergence free finite element space to a standard hybrid mixed poisson solver hence in each time step a hybrid mixed poisson solver is used which reflects the global nature of the incompressibility condition in the actual implementation of this fully discrete scheme the pressure field is also computed via the hybrid mixed poisson solver hence the scheme can be interpreted as a velocity pressure formulation that treats the incompressibility constraint and pressure forces implicitly but the viscous and convective part explicitly since we treat viscosity explicitly for the navier stokes equation our method shall be best suited for unsteady high reynolds number flows so that the cfl constraint is not too restrictive

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105