a robust nitsche s formulation for interface problems abstract in this work we propose a novel weighting for the interfacial consistency terms arising in a nitsche variational form we demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method consequently we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists an important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of a elements with arbitrarily small volume fractions b large material heterogeneities and c both large heterogeneities as well as arbitrarily small elements we then highlight the accuracy and efficiency of the proposed formulation through numerical examples focusing particular attention on interfacial quantities of interest

asymmetric simple exclusion process with open boundaries and askey wilson polynomials we study the one dimensional asymmetric simple exclusion process asep with open boundary conditions particles are injected and ejected at both boundaries it is clarified that the steady state of the model is intimately related to the askey wilson polynomials the partition function and the n point functions are obtained in the integral form with four boundary parameters the thermodynamic current is evaluated to confirm the conjectured phase diagram

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

numerical method for solving matrix coefficient elliptic equation with sharp edged interfaces solving elliptic equations with sharp edged interfaces is a challenging problem for most existing methods especially when the solution is highly oscillatory nonetheless it has wide applications in engineering and science an accurate and efficient method is desired we propose an efficient non traditional finite element method with non body fitting grids to solve the matrix coefficient elliptic equations with sharp edged interfaces extensive numerical experiments show that this method is second order accurate in the l norm and that it can handle both sharp edged interface and oscillatory solutions

finite size corrections to poisson approximations in general renewal success processes abstract consider a renewal process and let k 0 denote the random duration of a typical renewal cycle assume that on any renewal cycle a rare event called success can occur such successes lend themselves naturally to approximation by poisson point processes if each success occurs after a random delay however poisson convergence can be relatively slow because each success corresponds to a time interval not a point if k is an arithmetic variable a finite size correction fsc is known to speed poisson convergence by providing a second subdominant term in the appropriate asymptotic expansion this paper generalizes the fsc from arithmetic k to general k genomics applications require this generalization because they have already heuristically applied the fsc to p values involving absolutely continuous distributions the fsc also sharpens certain results in queuing theory insurance risk traffic flow and reliability theory

fundamental properties of infinite trees infinite trees naturally arise in the formalization and the study of the semantics of programming languages this paper investigates some of their combinatorial and algebraic properties that are especially relevant to semantics

simplifying subtyping constraints this paper offers a theoretical study of constraint simplification a fundamental issue for the designer of a practical type inference system with subtyping in the simpler case where constraints are equations a simple isomorphism between constrained type schemes and finite state automata yields a complete constraint simplification method using it as a guide for the intuition we move on to the case of subtyping and describe several simplification algorithms although no longer complete they are conceptually simple efficient and very effective in practice overall this paper gives a concise theoretical account of the techniques found at the core of our type inference system our study is restricted to the case where constraints are interpreted in a non structural lattice of regular terms nevertheless we highlight a small number of general ideas which explain our algorithms at a high level and may be applicable to a variety of other systems

solving elliptic problems with discontinuities on irregular domains the voronoi interface method we introduce a simple method dubbed the voronoi interface method to solve elliptic problems with discontinuities across the interface of irregular domains this method produces a linear system that is symmetric positive definite with only its right hand side affected by the jump conditions the solution and the solution s gradients are second order accurate and first order accurate respectively in the l norm even in the case of large ratios in the diffusion coefficient this approach is also applicable to arbitrary meshes additional degrees of freedom are placed close to the interface and a voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach both the locations of the additional degrees of freedom and their voronoi discretizations are straightforward in two and three spatial dimensions

approximate p values for local sequence alignments assume that two sequences from a finite alphabet are optimally aligned according to a scoring system that rewards similarities according to a general scoring scheme and penalizes gaps insertions and deletions under the assumption that the letters in each sequence are independent and identically distributed and the two sequences are also independent approximate p values are obtained for the optimal local alignment when either i there are at most a fixed number of gaps or ii the gap initiation cost is sufficiently large in the latter case the approximation can be written in the same form as the well known case of ungapped alignments

efficient inference of object types abstract m abadi and l cardelli have recently investigated a calculus of objects 1994 the calculus supports a key feature of object oriented languages an object can be emulated by another object that has more refined methods abadi and cardelli presented four first order type systems for the calculus the simplest one is based on finite types and no subtyping and the most powerful one has both recursive types and subtyping open until now is the question of type inference and in the presence of subtyping the absence of minimum typings poses practical problems for type inference in this paper we give an o n3 algorithm for each of the four type inference problems and we prove that all the problems are p complete we also indicate how to modify the algorithms to handle functions and records

estimating the gumbel scale parameter for local alignment of random sequences by importance sampling with stopping times the gapped local alignment score of two random sequences follows a gumbel distribution if computers could estimate the parameters of the gumbel distribution within one second the use of arbitrary alignment scoring schemes could increase the sensitivity of searching biological sequence databases over the web accordingly this article gives a novel equation for the scale parameter of the relevant gumbel distribution we speculate that the equation is exact although present numerical evidence is limited the equation involves ascending ladder variates in the global alignment of random sequences in global alignment simulations the ladder variates yield stopping times specifying random sequence lengths because of the random lengths and because our trial distribution for importance sampling occurs on a different sample space from our target distribution our study led to a mapping theorem which led naturally in turn to an efficient dynamic programming algorithm for the importance sampling weights numerical studies using several popular alignment scoring schemes then examined the efficiency and accuracy of the resulting simulations

a new finite element gradient recovery method superconvergence property this is the first in a series of papers in which a new gradient recovery method is introduced and analyzed it is proved that the method is superconvergent for translation invariant finite element spaces of any order the method maintains the simplicity efficiency and superconvergence properties of the zienkiewicz zhu patch recovery method in addition for uniform triangular meshes the method is superconvergent for the linear element under the chevron pattern and ultraconvergent at element edge centers for the quadratic element under the regular pattern applications of this new gradient recovery technique will be discussed in forthcoming papers

sampling rare events statistics of local sequence alignments a method to calculate probability distributions in regions where the events are very unlikely e g p approximately 10 40 is presented the basic idea is to map the underlying model on a physical system the system is simulated at a low temperature such that preferably configurations with originally low probabilities are generated since the distribution of such a physical system is known the original unbiased distribution can be obtained as an application local alignment of protein sequences is studied the deviation of the distribution p s of optimum scores from the extreme value distribution is quantified this deviation decreases with growing sequence length

score statistics of global sequence alignment from the energy distribution of a modified directed polymer and directed percolation problem sequence alignment is one of the most important bioinformatics tools for modern molecular biology the statistical characterization of gapped alignment scores has been a long standing problem in sequence alignment research using a variant of the directed path in random media model we investigate the score statistics of global sequence alignment taking into account in particular the compositional bias of the sequences compared such statistics are used to distinguish accidental similarity due to compositional similarity from biologically significant similarity to accommodate the compositional bias we introduce an extra parameter p indicating the probability for positive matching scores to occur when p is small a high scoring alignment obviously cannot come from compositional similarity when p is large the highest scoring point within a global alignment tends to be close to the end of both sequences in which case we say the system percolates by applying finite size scaling theory on percolating probability functions of various sizes sequence lengths the critical p at infinite size is obtained for alignment of length t the fact that the score fluctuation grows as chi t 1 3 is confirmed upon investigating the scaling form of the alignment score using the kolmogorov smirnov statistics test we show that the random variable if properly scaled follows the tracy widom distributions gaussian orthogonal ensemble for p slightly larger than pc and gaussian unitary ensemble for larger p although these results deepen our understanding of the distribution of alignment scores the use of these results in practical applications remains somewhat heuristic and needs to be further developed nevertheless the possibility of characterizing score statistics for modest system size sequence lengths via proper reparametrization of alignment scores is illustrated

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

a new weak galerkin finite element method for elliptic interface problems a new weak galerkin wg finite element method is introduced and analyzed in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces comparing with the existing wg algorithm for solving the same type problems the present wg method has a simpler variational formulation and fewer unknowns moreover the new wg algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders optimal order error estimates in both h 1 and l 2 norms are established for the present wg finite element solutions extensive numerical experiments have been conducted to examine the accuracy flexibility and robustness of the proposed wg interface approach in solving regular elliptic interface problems high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees moreover the wg method is shown to be able to accommodate very complicated interfaces due to its flexibility in choosing finite element partitions finally in dealing with challenging problems with low regularities the piecewise linear wg method is capable of delivering a second order of accuracy in l norm for both c 1 and h 2 continuous solutions

asymmetric inclusion process we introduce and explore the asymmetric inclusion process asip an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process asep in both processes random events cause particles to propagate unidirectionally along a one dimensional lattice of n sites in the asep particles are subject to exclusion interactions whereas in the asip particles are subject to inclusion interactions that coalesce them into inseparable clusters we study the dynamics of the asip derive evolution equations for the mean and probability generating function pgf of the sites occupancy vector obtain explicit results for the above mean at steady state and describe an iterative scheme for the computation of the pgf at steady state we further obtain explicit results for the load distribution in steady state with the load being the total number of particles present in all lattice sites finally we address the problem of load optimization and solve it under various criteria the asip model establishes bridges between statistical physics and queueing theory as it represents a tandem array of queueing systems with unlimited batch service and a tandem array of growth collapse processes

finite temperature local protein sequence alignment percolation and free energy distribution sequence alignment is a tool in bioinformatics that is used to find homological relationships in large molecular databases it can be mapped on the physical model of directed polymers in random media we consider the finite temperature version of local sequence alignment for proteins and study the transition between the linear phase and the biologically relevant logarithmic phase where the free energy grows linearly or logarithmically with the sequence length by means of numerical simulations and finite size scaling analysis we determine the phase diagram in the plane that is spanned by the gap costs and the temperature we use the most frequently used parameter set for protein alignment the critical exponents that describe the parameter driven transition are found to be explicitly temperature dependent furthermore we study the shape of the free energy distribution close to the transition by rare event simulations down to probabilities on the order 10 ensuremath 64 it is well known that in the logarithmic region the optimal score distribution t 0 is described by a modified gumbel distribution we confirm that this also applies for the free energy distribution tg0 however in the linear phase the distribution crosses over to a modified gaussian distribution

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

adaptive mesh refinement and superconvergence for two dimensional interface problems adaptive mesh refinement and the borgers algorithm are combined to generate a body fitted mesh which can resolve the interface with fine geometric details standard linear finite element method bas

asymmetric exclusion process and extremal statistics of random sequences a mapping is established between sequence alignment one of the most commonly used tools of computational biology at a certain choice of scoring parameters and the asymmetric exclusion process one of the few exactly solvable models of nonequilibrium physics the statistical significance of sequence alignments is characterized through studying the total hopping current of the discrete time and space version of the asymmetric exclusion process

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

on the fictitious domain and interpolation formulations of the matched interface and boundary mib method this work overcomes the difficulty of dealing with large curvatures in a high order matched interface and boundary mib method proposed for solving elliptic interface problems the mib method smoothly extends the solution across the interface so that standard high order central finite difference schemes can be used without the loss of accuracy one feature of the mib is that it disassociates the discretization of the elliptic equation from the enforcement of interface jump conditions the other is to make iterative use of only the lowest order jump conditions to determine the fictitious values on extended domains it is of arbitrarily high order in convergence in principle however its applicability was hindered by the lack of sufficiently many grid points to determine all the fictitious values required for high order schemes at the location where the curvature of the interface is relatively large we remove this obstacle by introducing a new concept the disassociation between the discretization and the domain extension we show that the improved mib method is robust for handling general irregular interfaces by extensive numerical experiments on the poisson equation and the helmholtz equation to better understand the mib method and other potential high order interface schemes we propose an alternative interpolation formulation of the mib method and show that the new formulation is essentially equivalent to the improved one

partially penalized immersed finite element methods for elliptic interface problems this article presents new immersed finite element ife methods for solving the popular second order elliptic interface problems on structured cartesian meshes even if the involved interfaces have nontrivial geometries these ife methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in ife functions with the enhanced stability due to the added penalty not only can these ife methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity but also numerical results indicate that their convergence rates in both the h 1 norm and the l 2 norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic ife methods in some situations trace inequalities are established for both linear and bilinear ife functions that are not only critical for the error analysis of these new ife methods but are also of a great potential to be useful in error analysis for other

weak galerkin methods for second order elliptic interface problems weak galerkin methods refer to general finite element methods for partial differential equations pdes in which differential operators are approximated by their weak forms as distributions such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions a weak galerkin finite element method wg fem is developed in this paper for solving elliptic pdes with discontinuous coefficients and interfaces theoretically it is proved that high order numerical schemes can be designed by using the wg fem with polynomials of high order on each element extensive numerical experiments have been carried out to validate the wg fem for solving second order elliptic interface problems high order of convergence is numerically confirmed in both l 2 and l norms for the piecewise linear wg fem special attention is paid to solve many interface problems in which the solution possesses a certain singularity due to the nonsmoothness of the interface a challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution the best known numerical scheme in the literature is of order o h to o h 1 5 for the solution itself in l norm it is demonstrated that the wg fem of the lowest order i e the piecewise constant wg fem is capable of delivering numerical approximations that are of order o h 1 75 to o h 2 in the l norm for c 1 or lipschitz continuous interfaces associated with a c 1 or h 2 continuous solution

occupancy correlations in the asymmetric simple inclusion process the asymmetric simple inclusion process asip a lattice gas model for unidirectional transport with irreversible aggregation has been proposed as an inclusion counterpart of the asymmetric simple exclusion process and as a batch service counterpart of the tandem jackson network to date the analytical tractability of the model has been limited while the average particle density in the model is easy to compute very little is known about the joint occupancy distribution to partially bridge this gap we study occupancy correlations in the asip we take an analytical approach to this problem and derive an exact formula for the covariance matrix of the steady state occupancy vector we verify the validity of this formula numerically in small asip systems where monte carlo simulations can provide reliable estimates for correlations in reasonable time and further use it to draw a comprehensive picture of spatial occupancy correlations in asip systems of arbitrary size

the asymmetric simple exclusion process an integrable model for non equilibrium statistical mechanics the asymmetric simple exclusion process asep plays the role of a paradigm in non equilibrium statistical mechanics we review exact results for the asep obtained by the bethe ansatz and put emphasis on the algebraic properties of this model the bethe equations for the eigenvalues of the markov matrix of the asep are derived from the algebraic bethe ansatz using these equations we explain how to calculate the spectral gap of the model and how global spectral properties such as the existence of multiplets can be predicted an extension of the bethe ansatz leads to an analytic expression for the large deviation function of the current in the asep that satisfies the gallavotti cohen relation finally we describe some variants of the asep that are also solvable by the bethe ansatz

accelerated convergence and robust asymptotic regression of the gumbel scale parameter for gapped sequence alignment searches through biological databases provide the primary motivation for studying sequence alignment statistics other motivations include physical models of annealing processes or mathematical similarities to e g first passage percolation and interacting particle systems here we investigate sequence alignment statistics partly to explore two general mathematical methods first we model the global alignment of random sequences heuristically with markov additive processes in sequence alignment the heuristic suggests a numerical acceleration scheme for simulating an important asymptotic parameter the gumbel scale parameter the heuristic might apply to similar mathematical theories second we extract the asymptotic parameter from simulation data with the statistical technique of robust regression robust regression is admirably suited to asymptotic regression and deserves to be better known for it

the extended finite element method for rigid particles in stokes flow a new method for the simulation of particulate flows based on the extended finite element method x fem is described in this method the particle surfaces need not conform to the finite element boundaries so that moving particles can be simulated without remeshing the near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh we present a weak form of the equations of motion useful for the simulation of freely moving particles and solve example problems for particles with prescribed and unknown velocities copyright 2001 john wiley sons ltd

the asymmetric exclusion process comparison of update procedures the asymmetric exclusion process asep has attracted a lot of interest not only because of its many applications e g in the context of the kinetics of biopolymerization and traffic flow theory but also because it is a paradigmatic model for nonequilibrium systems here we study the asep for different types of updates namely random sequential sequential sublattice parallel and parallel in order to compare the effects of the different update procedures on the properties of the stationary state we use large scale monte carlo simulations and analytical methods especially the so called matrix product ansatz mpa we present in detail the exact solution for the model with sublattice parallel and sequential updates using the mpa for the case of parallel update which is important for applications like traffic flow theory we determine the phase diagram the current and density profiles based on monte carlo simulations we furthermore suggest an mpa for that case and derive the corresponding matrix algebra

smoothly varying hopping rates in driven flow with exclusion we consider the one dimensional totally asymmetric simple exclusion process tasep with position dependent hopping rates the problem is solved in a mean field adiabatic approximation for a general smooth form of spatial rate variation numerical simulations of systems with hopping rates varying linearly against position constant rate gradient for both periodic and open boundary conditions provide detailed confirmation of theoretical predictions concerning steady state average density profiles and currents as well as open system phase boundaries to excellent numerical accuracy

universal large deviation function of the kardar parisi zhang equation in one dimension using the bethe ansatz we calculate the whole large deviation function of the displacement of particles in the asymmetric simple exclusion process asep on a ring when the size of the ring is large the central part of this large deviation function takes a scaling form independent of the density of particles we suggest that this scaling function found for the asep is universal and should be characteristic of all the systems described by the kardar parisi zhang equation in 1 1 dimension simulations done on two simple growth models are in reasonable agreement with this conjecture

stochastic dynamics of relativistic turbulence we apply generalized kolmogorov scaling to continuous time random walks coupled in space and time to obtain anomalous diffusion laws for relativistic turbulent media richardson s law for the mean square separation of two particles that are initially close together r t is recovered in the nonrelativistic limit while the ultrarelativistic limit is characterized by a different power law r t intermediate velocities are treated numerically showing a smooth transition from one regime to the other

accurate solution and gradient computation for elliptic interface problems with variable coefficients a new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface the main motivation is to get not only a second order accurate solution but also a second order accurate gradient from each side of the interface key to the new method is introducing the jump in the normal derivative of the solution as an augmented variable and rewriting the interface problem as a new pde that consists of a leading laplacian operator plus lower order derivative terms near the interface in this way the leading second order derivative jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling an upwind type discretization is used for the finite difference discretization at the irregular grid points on or near the interface so that the resulting coefficient matrix is an m matrix a multigrid solver is used to solve the linear system of equations and the gmres iterative me

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

solvable reaction diffusion processes without exclusion for reaction diffusion processes without exclusion in which the particles can exist in the same site of a one dimensional lattice we study all the integrable models which can be obtained by imposing a boundary condition on the master equation of the asymmetric diffusion process the annihilation process is also added the bethe ansatz solution and the exact n particle conditional probabilities are obtained

finite width model sequence comparison sequence comparison is a widely used computational technique in modern molecular biology in spite of the frequent use of sequence comparisons the important problem of assigning statistical significance to a given degree of similarity is still outstanding analytical approaches to filling this gap usually make use of an approximation that neglects certain correlations in the disorder underlying the sequence comparison algorithm here we use the longest common subsequence problem a prototype sequence comparison problem to analytically establish that this approximation does make a difference to certain sequence comparison statistics in the course of establishing this difference we develop a method that can systematically deal with these disorder correlations

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

a weak formulation for solving elliptic interface problems without body fitted grid a typical elliptic interface problem is casted as piecewise defined elliptic partial differential equations pde in different regions which are coupled together with interface conditions such as jumps in solution and flux across the interface in many situations such as the interface is moving the challenge is how to solve such a problem accurately robustly and efficiently without generating a body fitted mesh the key issue is how to capture complex geometry of the interface and jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the pde is not valid across the interface in this work we present a systematic formulation and further study of a second order accurate numerical method proposed in hou and liu 2005 16 for elliptic interface problem the key idea is to decompose the solution into two parts a singular part and a regular part the singular part captures the interface conditions while the regular part belongs to an appropriate space in the whole domain which can be solved by a standard finite element formulation in a general setup the two parts are coupled together we give an explicit study of the construction of the singular part and the discretized system for the regular part one key advantage of using weak formulation is that one can avoid assuming using more regularity than necessary of the solution and the interface we present the numerical algorithm and numerical tests in 3d to demonstrate the accuracy and other properties of our method

tableaux combinatorics for the asymmetric exclusion process and askey wilson polynomials introduced in the late 1960 s 33 43 the asymmetric exclusion process asep is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one dimensional lattice of n sites it has been cited as a model for traffic flow and protein synthesis in the most general form of the asep with open boundaries particles may enter and exit at the left with probabilities and and they may exit and enter at the right with probabilities and in the bulk the probability of hopping left is q times the probability of hopping right the first main result of this paper is a combinatorial formula for the stationary distribution of the asep with all parameters general in terms of a new class of tableaux which we call staircase tableaux this generalizes our previous work 14 15 for the asep with parameters 0 using our first result and also results of uchiyama sasamoto wadati 48 we derive our second main result a combinatorial formula for the moments of askey wilson polynomials since the early 1980 s there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials e g hermite charlier laguerre meixner however this is the first such formula for the askey wilson polynomials which are at the top of the hierarchy of classical orthogonal polynomials

occupation probabilities and fluctuations in the asymmetric simple inclusion process the asymmetric simple inclusion process asip a lattice gas model of unidirectional transport and aggregation was recently proposed as an inclusion counterpart of the asymmetric simple exclusion process in this paper we present an exact closed form expression for the probability that a given number of particles occupies a given set of consecutive lattice sites our results are expressed in terms of the entries of catalan s trapezoids number arrays which generalize catalan s numbers and catalan s triangle we further prove that the asip is asymptotically governed by the following i an inverse square root law of occupation ii a square root law of fluctuation and iii a rayleigh law for the distribution of interexit times the universality of these results is discussed

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

exact asymptotic results for the bernoulli matching model of sequence alignment finding analytically the statistics of the longest common subsequence lcs of a pair of random sequences drawn from c alphabets is a challenging problem in computational evolutionary biology we present exact asymptotic results for the distribution of the lcs in a simpler yet nontrivial variant of the original model called the bernoulli matching bm model we show that in the bm model for all c the distribution of the asymptotic length of the lcs suitably scaled is identical to the tracy widom distribution of the largest eigenvalue of a random matrix whose entries are drawn from a gaussian unitary ensemble

asymptotic behavior of a stationary silo with absorbing walls we study the nearest neighbors one dimensional uniform q model of force fluctuations in bead packs 1 a stochastic model to simulate the stress of granular media in two dimensional silos the vertical coordinate plays the role of time and the horizontal coordinate the role of space the process is a discrete time markov process with state space 1 n at each layer time the weight supported by each grain is a random variable of mean one its own weight plus the sum of random fractions of the weights supported by the nearest neighboring grains at the previous layer the fraction of the weight given to the right neighbor of the successive layer is a uniform random variable in 0 1 independent of everything the remaining weight is given to the left neighbor in the boundaries a uniform fraction of the weight leans on the wall of the silo this corresponds to absorbing boundary conditions for this model we show that there exists a unique invariant measure the mean weight at site i under the invariant measure is i n 1 i we prove that its variance is frac 1 2 i n 1 i 2 o n3 and the covariances between grains i j are of order o n3 moreover as n the law under the invariant measure of the weights divided by n2 around site integer part of rn r 0 1 converges to a product of gamma distributions with parameters 2 and 2 r 1 r 1 sum of two exponentials of mean r 1 r 2 liu et al 2 proved that for a silo with infinitely many weightless grains any product of gamma distributions with parameters 2 and 2 with 0 are invariant our result shows that as the silo grows the model selects exactly one of these gamma s at each macroscopic place

cutfem discretizing geometry and partial differential equations we discuss recent advances on robust unfitted finite element methods on cut meshes these methods are designed to facilitate computations on complex geometries obtained for example from computer

a nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity in this note we propose a finite element method for incompressible or compressible elasticity problems with discontinuous modulus of elasticity or if compressible poisson s ratio the problem is written on mixed form using p1 continuous displacements and elementwise p0 pressures leading to the possibility of eliminating the pressure beforehand in the compressible case in the incompressible case the method is augmented by a stabilization term penalizing the pressure jumps we show a priori error estimates under certain regularity hypothesis in particular we prove that if the exact solution is sufficiently smooth in each subdomain then the convergence order is optimal

dynamics and critical behavior of the q model the q model a random walk model rich in behavior and applications is investigated we introduce and motivate the q model via its application proposed by coppersmith et al to the flow of stress through granular matter at rest for a special value of its parameters the q model has a critical point that we analyze to characterize the critical point we imagine that a uniform load has been applied to the top of the granular medium and we study the evolution with depth of fluctuations in the distribution of load close to the critical point explicit calculation reveals that the evolution of load exhibits scaling behavior analogous to thermodynamic critical phenomena the critical behavior is remarkably tractable the harvest of analytic results includes scaling functions that describe the evolution of the variance of the load distribution close to the critical point and of the entire load distribution right at the critical point values of the associated critical exponents and determination of the upper critical dimension these results are of intrinsic interest as a tractable example of a random critical point of the many applications of the q model the critical behavior is particularly relevant to network models of river basins as we briefly discuss finally we discuss circumstances under which quantum network models that describe the surface electronic states of a quantum hall multilayer can be mapped onto the classical q model for mesoscopic multilayers of finite circumference the mapping fails instead a mapping to a ferromagnetic supersymmetric spin chain has proved fruitful we discuss aspects of the superspin mapping and give an elementary derivation of it making use of operator rather than functional methods

bethe ansatz calculation of the spectral gap of the asymmetric exclusion process we present a new derivation of the spectral gap of the totally asymmetric exclusion process on a half filled ring of size l by using the bethe ansatz we show that in the large l limit the bethe equations reduce to a simple transcendental equation involving the polylogarithm a classical special function by solving that equation the gap and the dynamical exponent are readily obtained our method can be extended to a system with an arbitrary density of particles

a posteriori error estimates based on the polynomial preserving recovery superconvergence of order o h 1 rho for some rho 0 is established for the gradient recovered with the polynomial preserving recovery ppr when the mesh is mildly structured consequently the ppr recovered gradient can be used in building an asymptotically exact a posteriori error estimator

gradient recovery for elliptic interface problem iii nitsche s method abstract this is the third paper on the study of gradient recovery for elliptic interface problems in our previous works guo and yang 2017 23 we developed gradient recovery methods for elliptic interface problems based on body fitted meshes and immersed finite element methods despite the efficiency and accuracy that these methods bring to recover the gradient there are still some cases in unfitted meshes where skinny triangles appear in the generated local body fitted triangulations that destroy the accuracy of recovered gradient near the interface in this paper we propose a gradient recovery technique based on nitsche s method for elliptic interface problems which avoids the loss of accuracy of gradient near the interface caused by skinny triangles we analyze the supercloseness between the gradient of the numerical solution by the nitsche s method and the gradient of the interpolation of the exact solution which leads to the superconvergence of the proposed gradient recovery method we also present several numerical examples to validate the theoretical results

a six vertex model as a diffusion problem derivation of correlation functions a cellular automaton which describes diffusion of particles with exclusion in one dimension is shown to be equivalent to a six vertex model on a critical line the arrow arrow correlation function of the six vertex model is calculated exactly on this line using a transfer matrix method

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients abstract assume that u x satisfies the problem l u x x i a i j u x j f x x u x 0 x in this article using interpolation postprocessing technique we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u x using piecewise polynomials of degrees bi k k 3 over a rectangular partition assume that k 3 is odd and x 0 is an interior vertex satisfying x 0 c using the new interpolation postprocessing formula presented in this study we show that the primal variable and the derivative of the post processed finite element solution using piecewise of degrees bi k k 3 at x 0 converge to the primal variable and the derivative of the exact solution with order o h k 3 ln h under suitable regularity and mesh conditions respectively finally we use numerical experiments to illustrate our theoretical findings

replica model for an unusual directed polymer in 1 1 dimensions and prediction of the extremal parameter of gapped sequence alignment statistics sequence alignment is one of the most important bioinformatics tools for modern molecular biology the statistical characterization of gapped alignment scores has been a long standing problem in sequence alignment research in this paper we provide a self contained exposition of sequence alignment a short review about how this problem is related to the directed polymer problem in statistical physics and some analytical results that can be used for predicting alignment score statistics basically we present two classes of solutions for the gapped alignment statistics by explicitly calculating the evolution of the few replica partition function in 1 1 dimensions we have obtained the conditions under which the more important extremal parameter lambda characterizing the alignment score statistics becomes predictable

interconvertibility of a class of set constraints and context free language reachability abstract we show the interconvertibility of context free language reachability problems and a class of set constraint problems given a context free language reachability problem we show how to construct a set constraint problem whose answer gives a solution to the reachability problem given a set constraint problem we show how to construct a context free language reachability problem whose answer gives a solution to the set constraint problem the interconvertibility of these two formalisms offers a conceptual advantage akin to the advantage gained from the interconvertibility of finite state automata and regular expressions in formal language theory namely a problem can be formulated in whichever formalism is most natural it also offers some insight into the o n 3 bottleneck for different types of program analysis problems and allows results previously obtained for context free language reachability problems to be applied to set constraint problems and vice versa

a second order virtual node method for elliptic problems with interfaces and irregular domains we present a second order accurate geometrically flexible and easy to implement method for solving the variable coefficient poisson equation with interfacial discontinuities or on irregular domains handling both cases with the same approach we discretize the equations using an embedded approach on a uniform cartesian grid employing virtual nodes at interfaces and boundaries a variational method is used to define numerical stencils near these special virtual nodes and a lagrange multiplier approach is used to enforce jump conditions and dirichlet boundary conditions our combination of these two aspects yields a symmetric positive definite discretization in the general case we obtain the standard 5 point stencil away from the interface for the specific case of interface problems with continuous coefficients we present a discontinuity removal technique that admits use of the standard 5 point finite difference stencil everywhere in the domain numerical experiments indicate second order accuracy in l

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

statistical self similarity of one dimensional growth processes abstract for one dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times we argue that for self similar growth from a single seed the universal distribution is the tracy widom distribution from the theory of random matrices and that for the growth from a flat substrate it is some other only numerically determined distribution in particular for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation from which the height distribution is identified as the tracy widom distribution

finite size scaling and universality for the totally asymmetric simple exclusion process the applicability of the concepts of finite size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one dimensional totally asymmetric simple exclusion process with open boundaries in the thermodynamic limit there are boundary induced transitions both of the first and second order between steady state phases of the model we derive finite size scaling expressions for the current near the continuous phase transition and for the local density near the first order transition under different stochastic dynamics and compare them to establish the existence of universal functions next we study numerically the finite size behavior of the lee yang zeros of the normalization factor for the different steady state probabilities

analysis of recovery type a posteriori error estimators for mildly structured grids some recovery type error estimators for linear finite elements are analyzed under o h 1 0 regular grids superconvergence of order o h 1 0 p is established for recovered gradients by three different methods as a consequence a posteriori error estimators based on those recovery methods are asymptotically exact

gradient recovery for elliptic interface problem ii immersed finite element methods abstract this is the second paper on the study of gradient recovery for elliptic interface problem in our previous work guo and yang 2016 17 we developed a novel gradient recovery technique for finite element method based on the body fitted mesh in this paper we propose new gradient recovery methods for two immersed interface finite element methods symmetric and consistent immersed finite method ji et al 2014 23 and petrov galerkin immersed finite element method hou et al 2004 22 and hou and liu 2005 20 compared to the body fitted mesh based gradient recovery method the new methods provide a uniform way of recovering gradient on regular meshes numerical examples are presented to confirm the superconvergence of both gradient recovery methods moreover they provide asymptotically exact a posteriori error estimators for both immersed finite element methods

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

objective method for estimating asymptotic parameters with an application to sequence alignment sequence alignment is an indispensable computational tool in modern molecular biology the model underlying biological sequence alignment is of interest to physicists because it approximates the statistical mechanics of dna and protein annealing while bearing an intimate relationship to models of directed polymers in random media recent methods for determining the statistics of random sequence alignments have reduced the computation time to less than 1 s opening up some interesting possibilities for online computation with biological search engines before implementation however the methods required an objective technique for computing regression coefficients pertinent to an asymptotic regime typically physicists estimate parameters pertinent to an asymptotic regime subjectively they eyeball their data estimate the asymptotic regime where the regression model holds with reasonable accuracy and then regress data only within the estimated asymptotic regime our publicly available computer program arrp replaces the subjective assessment of the asymptotic regime with an objective change point detection method increasing confidence in the scientific objectivity of the parameter estimates asymptotic regression has potential applications across most of physics

empirical null estimation using zero inflated discrete mixture distributions and its application to protein domain data in recent mutation studies analyses based on protein domain positions are gaining popularity over gene centric approaches since the latter have limitations in considering the functional context that the position of the mutation provides this presents a large scale simultaneous inference problem with hundreds of hypothesis tests to consider at the same time this article aims to select significant mutation counts while controlling a given level of type i error via false discovery rate fdr procedures one main assumption is that the mutation counts follow a zero inflated model in order to account for the true zeros in the count model and the excess zeros the class of models considered is the zero inflated generalized poisson zigp distribution furthermore we assumed that there exists a cut off value such that smaller counts than this value are generated from the null distribution we present several data dependent methods to determine the cut off value we also consider a two stage procedure based on screening process so that the number of mutations exceeding a certain value should be considered as significant mutations simulated and protein domain data sets are used to illustrate this procedure in estimation of the empirical null using a mixture of discrete distributions overall while maintaining control of the fdr the proposed two stage testing procedure has superior empirical power

asymptotically exact a posteriori error estimators part ii general unstructured grids in part i of this work siam j numer anal 41 2003 pp 2294 2312 we analyzed superconvergence for piecewise linear finite element approximations on triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms in this work we consider superconvergence for general unstructured but shape regular meshes we develop a postprocessing gradient recovery scheme for the finite element solution uh inspired in part by the smoothing iteration of the multigrid method this recovered gradient superconverges to the gradient of the true solution and becomes the basis of a global a posteriori error estimate that is often asymptotically exact next we use the superconvergent gradient to approximate the hessian matrix of the true solution and form local error indicators for adaptive meshing algorithms we provide several numerical examples illustrating the effectiveness of our procedures

universal distributions for growth processes in 1 1 dimensions and random matrices we develop a scaling theory for kardar parisi zhang growth in one dimension by a detailed study of the polynuclear growth model in particular we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape these distribution functions are computed using the partition function of gaussian random matrices in a cosine potential

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19

critical phenomena in sequence matching we give a generalization of the result of erdos and renyi on the length r n of the longest head run in the first n tosses of a coin consider two independent sequences x 1 x 2 cdots x m and y 1y 2 cdots y n suppose that x 1 x 2 cdots are i i d mu and y 1 y 2 cdots are i i d nu where mu and nu are possibly different distributions on a common finite alphabet s let p equiv p x 1 y 1 in 0 1 the length of the longest matching consecutive subsequence is m m n equiv max k x i r y j r for r 1 to k for some 0 leq i leq m k 0 leq j leq n k for m and n rightarrow infty with log m log mn rightarrow lambda in 0 1 our result is that there is a constant k equiv k mu nu lambda in 0 1 rbrack such that p lim m m n log 1 p mn k 1 the proof uses large deviation methods the constant k is determined from a variational formula involving the kullback liebler distance or relative entropy a simple necessary and sufficient condition for k 1 is given for the case m n lambda 1 2 and mu nu k 1 the set of mu nu lambda for which k 1 has nonempty interior the boundary of this set is the location of a phase transition the results generalize to more than two sequences and to markov chains a strong law of large numbers is given for the proportion of letters within the longest matching word the limiting proportion exhibits critical behavior similar to that of k

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process