a eulerian lagrangian numerical scheme for the dispersion convection equation using conjugate space time grids abstract a new numerical scheme is proposed for the dispersion convection equation which combines the utility of a fixed grid in eulerian coordinates with the computational power of the lagrangian method convection is formally decoupled from dispersion in a manner which does not leave room for ambiguity the resulting convection problem is solved by the method of characteristics on a grid fixed in space dispersion is handled by finite elements on a separate grid which may but need not coincide wit the former at selected points in spacetime information is projected from one grid to another by local interpolation the conjugate grid method is implemented by linear finite elements in conjunction with piecewise linear interpolation functions and applied to five problems ranging from predominant dispersion to pure convection the results are entirely free of oscillations numerical dispersion exists but can be brought under control either by reducing the spatial increment or by increasing the time step size of the grid used to solve the convection problem contrary to many other methods best results are often obtained when the courant number exceeds 1

atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

a monte carlo algorithm for lattice ribbons a lattice ribbon is a connected sequence of plaquettes subject to certain selfavoidance conditions the ribbon can be closed to form an object which is topologically either a cylinder or a mobius band depending on whether its surface is orientable or nonorientable we describe a grand canonical monte carlo algorithm for generating a sample of these ribbons prove that the associated markov chain is ergodic and present and discuss numerical results about the dimensions and entanglement complexity of the ribbons

generalized atmospheric sampling of self avoiding walks in this paper we introduce a new monte carlo method for sampling lattice self avoiding walks the method which we call gas generalized atmospheric sampling samples walks along weighted sequences by implementing elementary moves generated by the positive negative and neutral atmospheric statistics of the walks a realized sequence is weighted such that the average weight of states of length n is proportional to the number of self avoiding walks from the origin cn in addition the method also self tunes to sample from uniform distributions over walks of lengths in an interval 0 nmax we show how to implement gas using both generalized and endpoint atmospheres of walks and analyse our data to obtain estimates of the growth constant and entropic exponent of self avoiding walks in the square and cubic lattices

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

bfacf style algorithms for polygons in the body centered and face centered cubic lattices in this paper the elementary moves of the bfacf algorithm aragao de carvalho and caracciolo 1983 phys rev b 27 1635 45 aragao de carvalho and caracciolo 1983 nucl phys b 215 209 48 berg and foester 1981 phys lett b 106 323 6 for lattice polygons are generalized to elementary moves of bfacf style algorithms for lattice polygons in the body centered bcc and face centered fcc cubic lattices we prove that the ergodicity classes of these new elementary moves coincide with the knot types of unrooted polygons in the bcc and fcc lattices and so expand a similar result for the cubic lattice see janse van rensburg and whittington 1991 j phys a math gen 24 5553 67 implementations of these algorithms for knotted polygons using the gas algorithm produce estimates of the minimal length of knotted polygons in the bcc and fcc lattices

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

lattice stick number of knots the minimal number of straight line segments required to construct a polygonal presentation of the knot k in the cubic lattice is called the lattice stick number of the knot k denoted by it is known that if the crossing number of k satisfies and the main result of this paper is to improve this to if furthermore we will show that for and which implies that this lower bound cannot be improved except for knots with higher crossing numbers

isoparametric foliations and critical sets of eigenfunctions jakobson and nadirashvili j differ geom 53 1 177 182 1999 constructed a sequence of eigenfunctions on t 2 with a bounded number of critical points answering in the negative the question raised by yau problem section seminar on differential geometry princeton university press princeton 1982 which asks that whether the number of the critical points of eigenfunctions for the laplacian increases with the corresponding eigenvalues the present paper finds three interesting eigenfunctions on the minimal isoparametric hypersurface m n in s n 1 1 the corresponding eigenvalues are n 2n and 3n while their critical sets consist of 8 points a submanifold infinite many points and 8 points respectively on one of its focal submanifolds a similar phenomenon occurs

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

a characteristic mixed method with dynamic finite element space for convection dominated diffusion problems abstract a method for numerically solving convection dominated diffusion problems using a combination of the method of characteristics with the mixed finite element method is presented and analyzed in this paper different numbers of elements and different basis functions are implemented at different time levels in order to accurately resolve time changing critical features this new method provides the ability to obtain approximations of the solution and its gradient simultaneously adopt large time steps and make grid refinements and basis function adjustments at any time necessary convergence analysis and error estimates are established furthermore comments are made on the efficiency and capability of the method for accurately solving transient problems with localized phenomena such as fronts shocks and boundary layers

on the classification of isoparametric hypersurfaces with four distinct curvatures in spheres in this paper we give a new proof for the classification result in 3 we show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of clifford type provided that the multiplicities m 1 m 2 of the principal curvatures satisfy m 2 2m 1 1 this inequality is satisfied for all but five possible pairs m 1 m 2 with m 1 m 2 our proof implies that for m 1 m 2 1 1 the clifford system may be chosen in such a way that the associated quadratic forms vanish on the higher dimensional of the two focal manifolds for the remaining five possible pairs m 1 m 2 with m 1 m 2 see 13 1 and 15 this stronger form of our result is incorrect for the three pairs 3 4 6 9 and 7 8 there are examples of clifford type such that the associated quadratic forms necessarily vanish on the lower dimensional of the two focal manifolds and for the two pairs 2 2 and 4 5 there exist homogeneous examples that are not of clifford type cf 5 4 3 4 4

an iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media the miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure velocity equation and the concentration equation an iterative perturbation procedure is proposed and analyzed for the pressure velocity equation which is capable of producing as accurate a velocity approximation as the mixed finite element method and which requires the solution of symmetric positive definite linear systems only the velocity variable is involved in the linear systems and the pressure variable is obtained by substitution trivially applying perturbation methods can only give an error o epsilon while our iterative scheme can improve the error to o epsilon m at the m th iteration level where epsilon is a small positive number thus the convergence rate of our iterative procedure is o epsilon and consequently a small number of iterations is required theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

a new error analysis of characteristics mixed fems for miscible displacement in porous media the method of characteristics type is especially effective for convection dominated diffusion problems due to the nature of characteristic temporal discretization the method allows one to use a large time step in many practical computations while all previous theoretical analyses always required certain restrictions on the time stepsize here we present a new analysis to establish unconditionally optimal error estimates for a modified method of characteristics with a mixed finite element approximation to the miscible displacement problem in mathbb r d d 2 3 for this purpose we introduce a new characteristic time discrete system we prove that the l 2 error bound the characteristic time discrete systemof the fully discrete method of characteristics to the time discrete system is tau independent and the numerical solution is bounded in w 1 infty norm unconditionally with the boundedness optimal error estimates are established in a traditional manner numerical results confirm our th

statistical topology and knotting of fluctuating filaments abstract the aim of these notes is to provide an introduction to the topic of statistical topology with this name we refer to a combination of ideas and techniques from statistical mechanics and knot theory used to study the entanglement properties of fluctuating filaments some questions that we are going to address are the following i what is a knot and how can we identify it ii which is the probability of finding a random curve that is knotted iii how complex are these knots iv how big are they to try to partially answer these questions we will make use of few paradigmatic problems and try to investigate them by providing some state of the art theoretical and numerical techniques exercises and lists of open problems will be provided too

isoparametric foliation and yau conjecture on the first eigenvalue a well known conjecture of yau states that the first eigenvalue of every closed minimal hypersurface m n in the unit sphere s n 1 1 is just its dimension n the present paper shows that yau conjecture is true for minimal isoparametric hypersurfaces moreover the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non stable range

the first eigenvalue of the laplacian of an isoparametric minimal hypersurface in a unit sphere soit m n 1 une variete de riemann connexe compacte a n 1 dimensions sans bord et son laplacien agissant sur des fonctions lisses sur m a un spectre discret 0 0 m 1 m 2 m k m x soit f une immersion minimale de m n 1 dans la sphere unite a n dimensions s n 1 de l espace euclidien on etudie la premiere valeur propre du laplacien d une hypersurface isoparametrique minimale fermee dans s n 1

on the approximation of miscible displacement in porous media by a method of characteristics combined with a mixed method a miscible displacement of one incompressible fluid by another is modeled by a coupled system of partial differential equations the approximation of this system using a galerkin method that makes use of a modified method of characteristics for the concentration equation combined with a mixed finite element method for the pressure equation has been analyzed in comput math appl mech engrg 47 1984 pp 73 92 in this work we prove optimal error estimates in l 2 for this algorithm under milder restrictions in the time step than those required in the previous analysis in particular in the two dimensional case our estimates are valid when the time and space discretization parameters are of the same order which is a reasonable assumption in practice

on the universality of knot probability ratios let pn denote the number of self avoiding polygons of length n on a regular three dimensional lattice and let pn k be the number which have knot type k the probability that a random polygon of length n has knot type k is pn k pn and is known to decay exponentially with length sumners and whittington 1988 j phys a math gen 21 1689 94 pippenger 1989 discrete appl math 25 273 8 little is known rigorously about the asymptotics of pn k but there is substantial numerical evidence orlandini et al 1988 j phys a math gen 31 5953 67 marcone et al 2007 phys rev e 75 41105 rawdon et al 2008 macromolecules 41 4444 51 janse van rensburg and rechnitzer 2008 j phys a math theor 41 105002 that pn k grows as where nk is the number of prime components of the knot type k it is believed that the entropic exponent is universal while the exponential growth rate is independent of the knot type but varies with the lattice the amplitude ck depends on both the lattice and the knot type the above asymptotic form implies that the relative probability of a random polygon of length n having prime knot type k over prime knot type l is in the thermodynamic limit this probability ratio becomes an amplitude ratio it should be universal and depend only on the knot types k and l in this communication we examine the universality of these probability ratios for polygons in the simple cubic face centred cubic and body centred cubic lattices our results support the hypothesis that these are universal quantities for example we estimate that a long random polygon is approximately 28 times more likely to be a trefoil than be a figure eight independent of the underlying lattice giving an estimate of the intrinsic entropy associated with knot types in closed curves

bounds for the minimum step number of knots confined to slabs in the simple cubic lattice volume confinement is a key determinant of the topology and geometry of a polymer however the direct relationship between the two is not fully understood for instance recent experimental studies have constructed p4 cosmids i e p4 bacteriophages whose genome sequence and length have been artificially engineered and have shown that upon extraction their dna knot distribution differs from that of wild type bacteriophage p4 in particular it was observed that the complexity of the knots decreases sharply with the length of the packed genome this problem is the motivation of this paper here a polymer is modeled as a self avoiding polygon on the simple cubic lattice and the confining condition is such that the polygon is bounded between two parallel planes i e bounded within a slab we estimate the minimum length required for such a polygon to realize a knot type our numerical simulations show that in order to realize a prime knot with up to ten crossings in a 1 slab i e a slab of height 1 one needs a polygon of length strictly longer than the minimum length needed to realize the same knot when there is no confining condition in the case of the trefoil knot we can in fact establish this result analytically by proving that the minimum length required to tie a trefoil in the 1 slab is 26 which is greater than 24 the known minimum length required to tie a trefoil without a confinement condition additionally we find that in the 1 slab not all geometrical realizations of a given knot type are equivalent under bfacf moves this suggests that in certain confined volumes knowing the topology of a polymer is not enough to describe all its states

new monte carlo algorithms for interacting self avoiding walks in this paper we present two new monte carlo algorithms for simulating interacting self avoiding walks these algorithms are built on the berretti sokal and reptation algorithms for simulating non interacting self avoiding walks respectively by adding a metropolis step we enhance each algorithm by introducing special local moves in addition the reptation algorithm is further enhanced by running the simulations as a multiple markov chain each of these algorithms are studied here through an analysis of their autocorrelation properties in the case of the berretti sokal algorithm an increase in efficiency by a factor of around 40 is achieved for non interacting self avoiding walks

isoparametric hypersurfaces with g m 6 2 we prove that isoparametric hypersurfaces with g m 6 2 are homogeneous which answers dorfmeister neher s conjecture armatively and solves yau s problem in the case g 6

isoparametric foliations a problem of eells lemaire and conjectures of leung in this paper two sequences of minimal isoparametric hypersurfaces are constructed via representations of clifford algebras based on these we give estimates on eigenvalues of the laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres this improves results of z z tang and w j yan isoparametric foliation and yau conjecture on the first eigenvalue j differential geom 94 2013 521 540 z z tang y q xie and w j yan isoparametric foliation and yau conjecture on the first eigenvalue ii j funct anal 266 2014 6174 6199 eells and lemaire selected topics in harmonic maps c b m s regional conference series in mathematics 50 american mathematical society providence ri 1983 posed a problem to characterize the compact riemannian manifold m for which there is an eigenmap from m to sn as another application of our constructions the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres most importantly by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres we provide infinitely many counterexamples to two conjectures of leung minimal submanifolds in a sphere ii bull london math soc 23 1991 387 390 posed in 1991 on minimal submanifolds in unit spheres note that these conjectures of leung have been proved in the case that the normal connection is flat t hasanis and t vlachos ricci curvatures and minimal submanifolds pacific j math 197 2001 13 24

isoparametric hypersurfaces with four principal curvatures let m be an isoparametric hypersurface in the sphere s n with four distinct principal curvatures munzner showed that the four principal curvatures can have at most two distinct multiplicities m1 m 2 and stolz showed that the pair m1 m 2 must either be 2 2 4 5 or be equal to the multiplicities of an isoparametric hypersurface of fkm type constructed by ferus karcher and munzner from orthogonal representations of clifford algebras in this paper we prove that if the multiplicities satisfy m2 2m1 1 then the isoparametric hypersurface m must be of fkm type together with known results of takagi for the case m1 1 and ozeki and takeuchi for m1 2 this handles all possible pairs of multiplicities except for four cases for which the classification problem remains open

the weak galerkin method for solving the incompressible brinkman flow the brinkman equations are used to describe the dynamics of fluid flows in complex porous media with the high variability in the permeability coefficients which may take extremely large or small values this paper is devoted to the numerical analysis of a family of weak galerkin wg finite element methods for solving the time dependent brinkman problems this wg method is equipped with stable finite elements consisting of usual polynomials of degree k 1 for the velocity and polynomials of degree k 1 for the pressure the velocity element is enhanced by polynomials of degree k on the interface of the finite element partition all the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly defined spaces we further establish a priori error estimates in l 2 norm and h 1 norm and we provide a few numerical experiments to illustrate the behavior of the proposed scheme and confirm our theoretical findings regarding optimal convergence of the approximate solutions

unconditional convergence and optimal error estimates of a galerkin mixed fem for incompressible miscible flow in porous media in this paper we study the unconditional convergence and error estimates of a galerkin mixed fem with the linearized semi implicit euler scheme for the equations of incompressible miscible flow in porous media we prove that the optimal l 2 error estimates hold without any time step convergence conditions while all previous works require certain time step restrictions our theoretical results provide a new understanding on commonly used linearized schemes the proof is based on a splitting of the error into two parts the error from the time discretization of the pdes and the error from the finite element discretization of corresponding time discrete pdes the approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized semi implicit time discretizations

generalized atmospheric rosenbluth methods garm we show that the classical rosenbluth method for sampling self avoiding walks hammersley and morton 1954 j r stat soc b 16 23 rosenbluth and rosenbluth 1955 j chem phys 23 356 can be extended to a general algorithm for sampling many families of objects including self avoiding polygons the implementation relies on an elementary move which is a generalization of kinetic growth rather than only appending edges to the endpoint edges may be inserted at any vertex provided the resulting objects still lie within the same family this gives for the first time a kinetic growth algorithm for sampling self avoiding polygons we implement this method using pruning and enrichment grassberger 1997 phys rev e 56 3682 to sample self avoiding walks and polygons the algorithm can be further extended by mixing it with length preserving moves such as pivots and crank shaft moves

the bfacf algorithm and knotted polygons the bfacf algorithm applied to polygons involves sampling on a markov chain whose state space is the set of all polygons in three dimensions for the simple cubic lattice the authors prove that the ergodic classes of this markov chain are the knot classes of the polygons

isoparametric foliation and yau conjecture on the first eigenvalue ii abstract this is a continuation of 23 which investigated the first eigenvalues of minimal isoparametric hypersurfaces with g 4 distinct principal curvatures and focal submanifolds in unit spheres for the focal submanifolds with g 6 the present paper obtains estimates on all the eigenvalues among others giving an affirmative answer in one case to the problem posed in 23 which may be regarded as a generalization of yau s conjecture in two of the four unsettled cases in 23 for focal submanifolds m 1 of ot fkm type we prove the first eigenvalues to be their respective dimensions

ergodicity of the bfacf algorithm in three dimensions let omega be a walk in z3 let the endpoints of omega be fixed at x and gamma in z3 suppose that d x gamma maxi xi x x i gamma mod where x i3 x is the ith coordinate of the vertex x in z then the bfacf algorithm is ergodic for walks joining x and gamma provided that d x gamma 1

an h 1 galerkin mixed finite element method for parabolic partial differential equations in this paper an h1 galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts compared to the standard h1 galerkin procedure c1 continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method moreover it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method but without lbb consistency condition and quasiuniformity requirement on the finite element mesh finally a better rate of convergence for the flux in l2 norm is derived using a modified h1 galerkin mixed method in two and three space dimensions which confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed method under extra regularity assumptions on the exact solution

bounds for the minimum step number of knots in the simple cubic lattice knots are found in dna as well as in proteins and they have been shown to be good tools for structural analysis of these molecules an important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot here we address this problem by characterizing both analytically and numerically the minimum length also called minimum step number needed to form a particular knot in the simple cubic lattice our analytical work is based on improvement of a method introduced by diao to enumerate conformations of a given knot type for a fixed length this method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 which is 24 to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34 respectively using an independent method based on the bfacf algorithm we provide a complete list of numerical estimates upper bounds of the minimum step numbers for prime knots up to ten crossings which are improvements over current published numerical results we enumerate all minimum lattice knots of a given type and partition them into classes defined by bfacf type 0 moves

isoparametric foliations diffeomorphism groups and exotic smooth structures abstract in this paper we are concerned with interactions between isoparametric theory and differential topology two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves using differential topology we obtain several results towards the classification problem of isoparametric foliations up to equivalence in particular we show that each homotopy n sphere has the same isoparametric foliations as the standard sphere s n has except for n 4 reducing the classification problem on homotopy spheres to that on the standard sphere moreover we prove the uniqueness up to equivalence of isoparametric foliations with two points as the focal submanifolds on each sphere s n except for n 5 besides we show that the uniqueness holds on s 5 if and only if 0 diff s 4 z 2 i e pseudo isotopy implies isotopy for diffeomorphisms on s 4 at last some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds

flux corrected transport this paper describes a class of explicit eulerian finite difference algorithms for solving the continuity equation which are built around a technique called flux correction these flux corrected transport fct algorithms are of indeterminate order but yield realistic accurate results in addition to the mass conserving property of most conventional algorithms the fct algorithms strictly maintain the positivity of actual mass densities so steep gradients and inviscid shocks are handled particularly well this first paper concentrates on a simple one dimensional version of fct utilizing shasta a new transport algorithm for the continuity equation which is described in detail

an optimal order error estimate of the lowest order ellam mfem approximation to miscible displacement in three space dimensions abstract an optimal order error estimate for the lowest order ellam mfem time stepping procedure for the coupled system of time dependent nonlinear partial differential equations modeling miscible displacement that moves through the three dimensional subsurface porous media is proved under a practically reasonable time step constraint of t o h in the procedure the eulerian lagrangian localized adjoint method ellam and the mixed finite element method mfem are used to solve the transport equation and the pressure equation respectively this mathematically justifies the numerical advantages of the ellam mfem solution procedure

an optimal control least squares method for solving coupled flow transport systems modeling the transport of groundwater contamination leads generally to problems wherein the continuity equation is coupled with the contaminant transport equation and darcy s law such a system of equations is called the flow transport system and is difficult to solve numerically an optimal control h1 least squares formulation associated with a conjugate gradient algorithm is proposed for solving the coupled flow and transport problem at each time step each discretized differential equation is transformed into an optimal control problem this problem is then stated as an equivalent minimization problem the objective function associated with the transport problem is chosen in such a way that it accurately represents the advective behavior of the equation for high values of the peclet number the nazareth version of the conjugate gradient method is used for solving the problem iteratively at each iteration only the right hand side of the equation is changed thus the approach is attractive when large scale problems must be solved analysis of the preconditioning of the coefficient matrix provides insight into the behavior of this approach validation runs for a one dimensional example show accurate results for a wide range of peclet and courant numbers

convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics abstract a nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium conservation of mass for the mixture leads to an elliptic equation for pressure and conservation for the displacing fluid yields a convectiondominated parabolic equation for the concentration of that fluid a sequential implicit time stepping procedure is defined in which the pressure and darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a galerkin finite element method and the method of characteristics optimal order convergence in l2 is proved time truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the concentration equation temporal and spatial errors are lessened by direct computation of the velocity in the mixed method as opposed to differentiation of the pressure several extensions of these results are outlined

asymptotics of knotted lattice polygons we use monte carlo methods to investigate the asymptotic behaviour of the number and mean square radius of gyration of polygons in the simple cubic lattice with fixed knot type let be the number of n edge polygons of a fixed knot type in the cubic lattice and let be the mean square radius of gyration of all the polygons counted by if we assume that where is the growth constant of polygons of knot type and is the entropic exponent of polygons of knot type then our numerical data are consistent with the relation where is the unknot and is the number of prime factors of the knot if we assume that then our data are consistent with both and being independent of these results support the claims made in janse van rensburg and whittington 1991a 24 3935 and orlandini et al 1996 29 l299 1998 topology and geometry in polymer science ima volumes in mathematics and its applications berlin springer