local discontinuous galerkin methods for the stokes system in this paper we introduce and analyze local discontinuous galerkin methods for the stokes system for a class of shape regular meshes with hanging nodes we derive a priori estimates for the l2 norm of the errors in the velocities and the pressure we show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure for any k ge1 we also consider the case in which all the unknowns are approximated with polynomials of degree k and show that although the orders of convergence remain the same the method is more efficient numerical experiments verifying these facts are displayed

two dimensional lattice vesicles and polygons we consider a lattice model of two dimensional vesicles in which the boundary of the vesicle is the perimeter of a self avoiding polygon embeddable in the square lattice with fixed boundary length m we incorporate an osmotic pressure difference by associating a fugacity with the area enclosed by the polygon we derive rigorous results concerning the behaviour of the associated free energy and the form of the phase diagram by deriving exact values ofthe numbers of polygons with m edges which enclose area n and analysing the resulting series we obtain the free energy the phase boundary and various scaling exponents and amplitudes numerically

plane partition vesicles we examine partitions and their natural three dimensional generalizations plane partitions as models of vesicles undergoing an inflation deflation transition the phase diagrams of these models include a critical point corresponding to an inflation deflation transition and exhibits multicritical scaling in the vicinity of a multicritical point located elsewhere on the critical curve we determine the locations of the multicritical points by analysing the generating functions using analytic and numerical means in addition we determine the numerical values of the multicritical scaling exponents associated with the multicritical scaling regimes in these models

statistics of lattice animals the authors investigate the large n behaviour of the number of lattice animals with n vertices having alpha cycles per vertex they prove concavity and continuity properties for the corresponding growth constant and in particular show that lattice trees are exponentially scarce in the set of lattice animals they also consider the corresponding generating function and prove a number of theorems which bound it and set other limits on its possible behaviour

the topology of interfaces a discrete model of an interface separating two phases is considered the interface is defined as the boundary of a self avoiding n omino with volume n embedded in the hypercubic lattice the author proves that if there is no surface tension in the interface then it has non trivial topology with probability one in the scaling limit here non trivial topology means that the interface will consist of several disjoint components and will have non zero genus moreover the total area of the n omino will be proportional to its volume

monte carlo study of self avoiding surfaces two models of self avoiding surfaces on the cubic lattice are studied by monte carlo simulations both the first model with fluctuating boundary and the second one with a fixed boundary are found to belong to the universality class of branched polymers the algorithms as well as the methods used to extract the critical exponents are described in detail the results are compared to other recent estimates in the literature

self avoiding surfaces with knotted boundaries the authors consider orientable self avoiding surfaces with genus g embedded in the hypercubic lattice they consider a surface which has a boundary component and non zero genus and devise a construction which will reduce the genus of the surface this result enables them to study embeddings of surfaces in the three dimensional lattice where a surface of genus g may have a boundary component which is a knot of genus g or g they prove that the growth constants of these surfaces are independent of the knot type of the boundary component and they derive inequalities between the associated critical exponents

self avoiding surfaces the set of self avoiding random surfaces sn h with n plaquettes and h boundary components is considered the concatenation of the surfaces in sn h and new constructions which either increase or decrease the number of boundary components of a surface are studied these constructions make it possible to prove the existence of growth constants beta h for the cardinality sn h of sn h for each h or 1 in two dimensions and h or 0 in d or 3 dimensions the authors prove that beta h beta 1 for all h or 1 in d or 2 dimensions in addition they prove that beta 0 or 3 dimensions and that in two dimensions beta 1 beta where beta is the growth constant of the set sn the set of all self avoiding surfaces in two dimensions finally by postulating the existence of a critical exponent phi h for each set sn h by assuming that sn h approximately n phi h beta nh they derive bounds on phi h from the constructions defined on the surface in sn h

self avoiding random surfaces monte carlo study using oct tree data structure self avoiding random surfaces on a cubic lattice are studied by extensive monte carlo sampling the surfaces have an empty boundary and the topology of a 2 sphere an oct tree data structure allows good statistics to be obtained for surfaces whose plaquette number is up to an order of magnitude greater than in previous investigations the new simulation strategy is explained in detail and compared with previous ones the critical plaquette fugacity mu 1 and the entropic exponent theta are determined by maximum likelihood methods and by logarithmic plots of the average surface area versus fugacity the latter approach which produces results having much better convergence by taking advantage of the scaling properties of several runs at various fugacities leads to the estimates mu 1 729 or 0 036 and theta 1 500 or 0 026 linear regression estimates for the radius of gyration exponent give v 0 509 or 0 004 while the asymptotic ratio of surface area over average volume enclosed approaches a finite value 3 18 or 0 03 the results give strong corroborating evidence that this long controversial problem belongs to the universality class of branched polymers

a discontinuous galerkin method with nonoverlapping domain decomposition for the stokes and navier stokes problems a family of discontinuous galerkin finite element methods is formulated and analyzed for stokes and navier stokes problems an inf sup condition is established as well as optimal energy estimates for the velocity and l 2 estimates for the pressure in addition it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces

critical exponents for lattice animals with fixed cyclomatic index the authors derive an inequality between the number of trees and the number of lattice animals with exactly c cycles an c for all positive c if they assume that an c approximately n theta c lambda nc n to infinity c fixed they use this to show that theta c theta o c where theta 0 is the corresponding exponent for trees

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

bound ratio minimization of filter bank frames this paper investigates and solves the problem of frame bound ratio minimization for oversampled perfect reconstruction pr filter banks fbs for a given analysis prfb a finite dimensional convex optimization algorithm is derived to redesign the subband gain of each channel the redesign minimizes the frame bound ratio of the fb while maintaining its original properties and performance the obtained solution is precise without involving frequency domain approximation and can be applied to many practical problems in signal processing the optimal solution is applied to subband noise suppression and tree structured fb gain optimization resulting in deeper insights and novel solutions to these two general classes of problems and considerable performance improvement effectiveness of the optimal solution is demonstrated by extensive numerical examples

surfaces in the hypercubic lattice let sn h g be the number of surfaces with h boundary components and with genus g embedded in the hypercubic lattice then the author proves that for all h or 0 and g or 0 there exists a constant c6 0 in two dimensions such that o e c6 log n 2 beta 1 0n 0 and k3 0 such that in three and more dimensions o e k7 logn 3 beta 1 0n or phi 1 0 2h 4g in three and more dimensions

counting closed 2 manifolds in tubes in hypercubic lattices we consider closed 2 manifolds 2 manifolds without boundary embedded in tubes in the hypercubic lattice z d for orientable 2 manifolds with fixed genus we prove that the exponential growth rate is independent of the genus and we use this to prove a pattern theorem for manifolds with fixed genus we prove a similar theorem for the non orientable case for if the genus is not fixed then we prove a pattern theorem and use this to show that 2 manifolds with genus less than any fixed number are exponentially rare so the typical genus increases with the size of the manifold in four and higher dimensions we prove that orientable manifolds are exponentially rare and are dominated by non orientable manifolds in four dimensions all except exponentially few 2 manifolds both orientable and non orientable contain a local knotted ball pair

polygonal polyominoes on the square lattice we study a proper subset of polyominoes called polygonal polyominoes which are defined to be self avoiding polygons containing any number of holes each of which is a self avoiding polygon the staircase polygon subset with staircase holes is also discussed the internal holes have no common vertices with each other nor any common vertices with the surrounding polygon there are no holes within holes we use the finite lattice method to count the number of polygonal polyominoes on the square lattice series have been derived for both the perimeter and area generating functions it is known that while the critical point is unchanged by a finite number of holes when the number of holes is unrestricted the critical point changes the area generating function coefficients grow exponentially with a growth constant greater than that for polygons with a finite number of holes but less than that of polyominoes we provide an estimate for this growth constant and prove that it is strictly less than that for polyominoes also we prove that enumerating by perimeter the generating function of polygonal polyominoes has zero radius of convergence and furthermore we calculate the dominant asymptotics of its coefficients using rigorous bounds

the local discontinuous galerkin method for the oseen equations we introduce and analyze the local discontinuous galerkin method for the oseen equations of incompressible fluid flow for a class of shape regular meshes with hanging nodes we derive optimal a priori estimates for the errors in the velocity and the pressure in l 2 and negative order norms numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of reynolds numbers

on the universality class of planar self avoiding surfaces with fixed boundary using a modified version of a monte carlo algorithm proposed by sterling and greensite 1983 the authors obtain the exponents theta 1 51 or 0 25 and nu 0 502 or 0 024 for planar self avoiding surfaces with a fixed boundary in three dimensions consistent with the conjectured exact values for branched polymers they show how the modifications are needed to obtain a viable distribution of surfaces

mixed hp dgfem for incompressible flows we consider several mixed discontinuous galerkin approximations of the stokes problem and propose an abstract framework for their analysis using this framework we derive a priori error estimates for hp approximations on tensor product meshes we also prove a new stability estimate for the discrete divergence bilinear form