simulating the dynamics of inextensible vesicles by the penalty immersed boundary method in this paper we develop an immersed boundary ib method to simulate the dynamics of inextensible vesicles interacting with an incompressible fluid in order to take into account the inextensibility constraint of the vesicle the penalty immersed boundary pib method is used to virtually decouple the fluid and vesicle dynamics as numerical tests of our current pib method the dynamics of single and multiple inextensible vesicles under shear flows have been extensively explored and compared with the previous literature the method is also validated by a series of convergence study which confirms its consistent first order accuracy on the velocity field the vesicle configuration the vesicle area and the perimeter errors in addition the method is also applied to study a binary component vesicle problem

reductions of binary trees and lattice paths induced by the register function abstract the register function or horton strahler number of a binary tree is a well known combinatorial parameter we study a reduction procedure for binary trees which offers a new interpretation for the register function as the maximal number of reductions that can be applied to a given tree in particular the precise asymptotic behavior of the number of certain substructures branches that occur when reducing a tree repeatedly is determined in the same manner we introduce a reduction for simple two dimensional lattice paths from which a complexity measure similar to the register function can be derived we analyze this quantity as well as the cumulative size of an iteratively reduced lattice path asymptotically

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

the martin boundary in non lipschitz domains the martin boundary with respect to the laplacian and with respect to uniformly elliptic operators in divergence form can be identified with the euclidean boundary in c domains where x bx loglog 1 x logloglog 1 x b small a counterexample shows that this result is very nearly sharp

register allocation for unary binary trees we study the number of registers required for evaluating arithmetic expressions formed with any set of unary and binary operators our approach consists in a singularity analysis of intervening generating functions combined with a use of complex mellin inversion we illustrate it first by rederiving the known results about binary trees and then extend it to the fully general case of unary binary trees the method used as mentioned in the conclusion is applicable to a wide class of combinatorial sums

uniqueness for diffusions with piecewise constant coefficients letl be a second order partial differential operator inrd letrd be the finite union of disjoint polyhedra suppose that the diffusion matrix is everywhere non singular and constant on each polyhedron and that the drift coefficient is bounded and measurable we show that the martingale problem associated withl is well posed

intrinsic ultracontractivity and the dirichlet laplacian we show that if d x y 0 x 1 f x y 1 where f is negative upper semicontinuous and bounded below then the semigroup associated with the dirichlet laplacian on d is intrinsically ultracontractive analogs in higher dimensions are proved we construct domains of infinite area with this property

the lifetime of conditioned brownian motion in certain lipschitz domains for certain lipschitz domains d we obtain a series expansion for the distribution of the lifetime d of conditioned brownian motion on d from this we determine mathop lim limits t to infty text t 1 text log text p x h tau d text t mathop lim limits t to infty text t 1 text log text p x tau d text t text lambda d text where d is the first eigenvalue of 1 2 on d

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

lifetimes of conditioned diffusions we investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non divergence form can be found the critical value of the parameter is found for each of the following classes of domains l p domains p n 1 uniformly regular twistedl p domains p n 1 and twisted holder domains 1 3 a related parabolic boundary harnack principle is proved

a boundary integral method for computing the dynamics of an epitaxial island in this paper we present a boundary integral method for computing the quasi steady evolution of an epitaxial island the problem consists of an adatom diffusion equation with desorption on terrace and a kinetic boundary condition at the step island boundary the normal velocity for step motion is determined by a two sided flux the integral formulation of the problem involves both double and single layer potentials due to the kinetic boundary condition numerical tests on a growing shrinking circular or a slightly perturbed circular island are in excellent agreement with the linear analysis demonstrating that the method is stable efficient and spectrally accurate in space nonlinear simulations for the growth of perturbed circular islands show that sharp tips and flat edges will form during growth instead of the usual dense branching morphology seen throughout physical and biological systems driven out of equilibrium in particular bales zangwill instability is manifested in the form of wave like fronts meandering instability around the tip regions the numerical techniques presented here can be applied generally to a class of free moving boundary problems in physical and biological science

doob s conditioned diffusions and their lifetimes on utilise les resultats de donsker et varadhan pour montrer que la queue de la distribution de la duree de vie decroit exponentiellement

nonlinear simulation of shape preserving delta growth abstract in this paper we demonstrate the existence of shape preserving self similar delta we focus our study on a gilbert type delta growing by the sediment discharged at the river mouth the evolving velocity of the shoreline depends on its local geometrical configuration including the water depth repose angle and curvature linear analysis of this local model suggests that when the sediment supply to the shoreline front can be maintained at a constant value unstable growth occurs for adverse bathymetry back tilted basement and there exists a critical water depth that leads to a self similar evolution a novel nonlinear analysis based on a rescaling idea reveals that there exists a critical flux j c at the shoreline such that a desired shoreline shape can be achieved and maintained self similarly independent of the water depth information though the critical flux j c may depend on the water depth we then develop a semi implicit numerical method to investigate the nonlinear dynamic of the shoreline our numerical results are in excellent agreement with the linear theory when the shape perturbation is small and confirm that in the nonlinear regime the shoreline may evolve self similarly under j c in particular we demonstrate that a prescribed shoreline morphology can be achieved by a well designed j c the existence of shape preserving growing delta goes beyond the well known dynamical patterns and highlights the feasibility of shape control

the lifetime of conditioned brownian motion in planar domains of infinite area in this paper it is shown that all expected lifetimes ofh processes ind are finite if and only if the area ofd is finite ifd x y o x y o x x where o x o are two lipschitz functions we show that if is a bounded convex region in the plane there is anh process in with expected lifetime at leastc area we also give an example of a planar domaind of infinite area such that the expected lifetime of eachh process ind is finite

nonlinear simulations of elastic fingering in a hele shaw cell this work is motivated by the recent experiments of two reacting fluids in a hele shaw cell podgorski et al 2007 and associated linear stability analysis of a curvature weakening model he et al 2012 unlike the classical hele shaw problem posed for moving interfaces with surface tension the curvature weakening model is concerned with a newly produced gel like phase that stiffens the interface thus the interface is modeled as an elastic membrane with curvature dependent rigidity that reflects geometrically induced breaking of intermolecular bonds here we are interested in exploring the long time interface dynamics in the nonlinear regime we perform simulations using a spectrally accurate boundary integral method together with a rescaling scheme to dramatically speed up the intrinsically slow evolution of the interface we find curvature weakening inhibits tip splitting and promotes side branching morphology at long times numerical results reveal that there exist nonlinear stable self similarly evolving morphologies

the h path distribution of the lifetime of conditioned brownian motion for non smooth domains consider the h paths of doob and let d be the lifetime of brownian motion killed at d where d is a bounded lipschitz or nta domain in n then log p x h tau d t sim lambda d t as t to infty where d is the first positive eigenvalue of 1 2 on d

topological self similarity on the random binary tree model asymptotic analysis on some statistical properties of the random binary tree model is developed we quantify a hierarchical structure of branching patterns based on the horton strahler analysis we introduce a transformation of a binary tree and derive a recursive equation about branch orders as an application of the analysis topological self similarity and its generalization is proved in an asymptotic sense also some important examples are presented

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

holder domains and the boundary harnack principle the constant c depends on l only through the constant cl defined in 1 below of course c also depends on d v and k the above result shows that the boundary harnack principle holds in all holder domains it was first proven for the range 1 2 1 by bass and burdzy 1990a theorems 3 5 and 4 5 it was subsequently extended by banuelos unpublished to the full range of provided the domain also satisfies a uniform capacity condition on the boundary in banuelos 1990 a class of domains called uniformly holder domains of order was introduced a boundary harnack principle may be obtained for such domains for all 0 1 by a variation of the proof of theorem 1 in order to make this note compact we refer

on a problem of yekutieli and mandelbrot about the bifurcation ratio of binary trees concerning the horton strahler number or register function of binary trees yekutieli and mandelbrot posed the problem of analyzing the bifurcation ratio of the root which means how many maximal subtrees of register function one less than the whole tree are present in the tree we show that if all binary trees of size n are considered to be equally likely then the average value of this number of subtrees is asymptotic to 3 341266 log4 n where an analytic expression for the numerical constant is available and x is a small periodic function of period 1 which is also given explicitly additionally we sketch the computation of the variance and also of higher bifurcation ratios

on an estimate of cranston and mcconnell for elliptic diffusions in uniform domains we show that ifd n n 3 n 3 is a bounded uniform domain then the lifetime of the doobh paths ind for elliptic diffusions in divergence form is finite this result holds for any bounded domaind in the plane

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

lifetime of conditioned brownian motion in lipschitz domains if d d d 3 is bounded and has lipschitz boundary then the expected lifetime of any brownian h path process in d is finite

a boundary harnack principle in twisted holder domains the boundary harnack principle for the ratio of positive harmonic functions is shown to hold in twisted holder domains of order a for a e 1 2 1 for each a e 0 1 2 there exists a twisted h6lder domain of order a for which the boundary harnack principle fails extensions are given to l harmonic

a rescaling scheme with application to the long time simulation of viscous fingering in a hele shaw cell in this paper we present a time and space rescaling scheme for the computation of moving interface problems the idea is to map time space such that the interfaces can evolve exponentially fast in the new time scale while the area volume enclosed by the interface remains unchanged the rescaling scheme significantly reduces the computation time especially for slow growth and enables one to accurately simulate the very long time dynamics of moving interfaces we then implement this scheme in a hele shaw problem examine the dynamics for a number of different injection fluxes and present the largest and most pronounced viscous fingering simulations to date

a phase field approach in the numerical study of the elastic bending energy for vesicle membranes in this paper we compute the equilibrium configurations of a vesicle membrane under elastic bending energy with prescribed volume and surface area a variational phase field method is developed for such a problem discrete finite difference approximations and numerical simulations are carried out in the axial symmetrical cases different energetic bifurcation phenomena are discussed

central limit theorem for the bifurcation ratio of a random binary tree in order to formulate and examine the central limit theorem for a binary tree numerically a method for generating random binary trees is presented we first propose the correspondence between binary trees and a certain type of binary sequences which we call dyck sequences then the method for generating random dyck sequences is shown also we propose the method of branch ordering of a binary tree by means of only the corresponding dyck sequence we confirm that the method is in good consistency with the topological analysis of binary trees known as the horton strahler analysis two types of central limit theorem are numerically confirmed and the obtained results are expressed in simple forms furthermore the proposed method is available for a wide range of the topological analysis of binary trees