exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

transfer matrices for the totally asymmetric simple exclusion process we consider the totally asymmetric simple exclusion process on a finite lattice with open boundaries we show using the recursive structure of the markov matrix that encodes the dynamics that there exist two transfer matrices tl 1 l and that intertwine the markov matrices of consecutive system sizes this semi conjugation property of the dynamics provides an algebraic counterpart for the matrix product representation of the steady state of the process

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

stationary distributions of multi type totally asymmetric exclusion processes we consider totally asymmetric simple exclusion processes with n types of particle and holes n taseps on z and on the cycle z n angel recently gave an elegant construction of the stationary measures for the 2 tasep based on a pair of independent product measures we show that angel s construction can be interpreted in terms of the operation of a discrete time m m 1 queueing server the two product measures correspond to the arrival and service processes of the queue we extend this construction to represent the stationary measures of an n tasep in terms of a system of queues in tandem the proof of stationarity involves a system of n 1 taseps whose evolutions are coupled but whose distributions at any fixed time are independent using the queueing representation we give quantitative results for stationary probabilities of states of the n tasep on z n and simple proofs of various independence and regeneration properties for systems on z

a non overlapping domain decomposition method with non matching grids for modeling large finite antenna arrays a non overlapping domain decomposition method ddm is proposed herein to solve maxwell equations in r 3 in this work the maxwell equations are discretized using a vector finite element method with hierarchical h curl vector basis functions there are two major ingredients in the proposed non overlapping ddm a a proper 1st order transmission condition to enforce field continuity across domain boundaries and b a cement technique to allow non matching grids for neighboring domains moreover a detail fourier analysis of the transmission condition for a canonical half space example is presented the analysis provides significant insights into the convergence behavior of the proposed non overlapping ddm for solving electromagnetic radiation problems such as the large finite antenna arrays particularly for the antenna arrays the proposed non overlapping ddm is extremely efficient since the formulation can easily incorporate geometrical repetitions exponentially tapered notch vivaldi antenna arrays with size up to 100x100 elements are solved on a common pc to validate the proposed non overlapping ddm

exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra we study the partially asymmetric exclusion process with open boundaries we generalize the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and currents valid for all values of the asymmetry parameter q due to the relationship between the matrix algebra and the q deformed quantum harmonic oscillator algebra we find that q hermite polynomials along with their orthogonality properties and generating functions are of great utility we employ two distinct sets of q hermite polynomials one for q 1 it turns out that these correspond to two distinct regimes the previously studied case of forward bias q 1 where the boundaries support a current opposite in direction to the bulk bias for the forward bias case we confirm the previously proposed phase diagram whereas the case of reverse bias produces a new phase in which the current decreases exponentially with system size

matrix product representation of the stationary state of the open zero range process many one dimensional lattice particle models with open boundaries like the paradigmatic asymmetric simple exclusion process asep have their stationary states represented in the form of a matrix product with matrices that do not explicitly depend on the lattice site in contrast the stationary state of the open 1d zero range process zrp takes an inhomogeneous factorized form with site dependent probability weights we show that in spite of the absence of correlations the stationary state of the open zrp can also be represented in a matrix product form where the matrices are site independent non commuting and determined from algebraic relations resulting from the master equation we recover the known distribution of the open zrp in two different ways first using an explicit representation of the matrices and boundary vectors second from the sole knowledge of the algebraic relations satisfied by these matrices and vectors finally an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden markov chains

one dimensional partially asymmetric simple exclusion process with open boundaries orthogonal polynomials approach the stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered the so called matrix product ansatz is employed this enables us to construct the stationary state in the form of the product of the matrices d and e noticing the fact that the matrix c d e for the model is related to certain q orthogonal polynomials the model is analysed for a wide range of parameters the current and the correlation length are evaluated in the thermodynamic limit it turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case

multiline queues with spectral parameters using the description of multiline queues as functions on words we introduce the notion of a spectral weight of a word by defining a new weighting on multiline queues we show that the spectral weight of a word is invariant under a natural action of the symmetric group giving a proof of the commutativity conjecture of arita ayyer mallick and prolhac we give a determinant formula for the spectral weight of a word which gives a proof of a conjecture of the first author and linusson

an exclusion process on a tree with constant aggregate hopping rate we introduce a model of a totally asymmetric simple exclusion process tasep on a tree network where the aggregate hopping rate is constant from level to level with this choice for hopping rates the model shows the same phase diagram as the one dimensional case the potential applications of our model are in the area of distribution networks where a single large source supplies material to a large number of small sinks via a hierarchical network we show that mean field theory mft for our model is identical to that of the one dimensional tasep and that this mft is exact for the tasep on a tree in the limit of large branching ratio b or equivalently large coordination number we then present an exact solution for the two level tree or star network that allows the computation of any correlation function and confirm how mean field results are recovered as b as an example we compute the steady state current as a function of branching ratio we present simulation results that confirm these results and indicate that the convergence to mft with large branching ratio is quite rapid

totally asymmetric exclusion process on a ring with internal degrees of freedom a totally asymmetric exclusion process on a ring with v nonconserved internal degrees of freedom where particles hop forward with a rate that depends on their internal state has been studied we show using a mapping of the model to a zero range process with v different kinds of boxes that steady state weights can be written in a matrix product form and calculate the spatial correlations exactly a comparison of the model with an equivalent conserved system reveals that unequal hopping rates of particles belonging to different internal states are responsible for the nontrivial correlations

multispecies tasep and the tetrahedron equation we introduce a family of layer to layer transfer matrices in a three dimensional 3d lattice model which can be viewed as partition functions of the q oscillator valued six vertex model on an m n square lattice by invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions at q 0 and m n they ultimately yield a new proof of the steady state formula for the n species totally asymmetric simple exclusion process obtained recently by the authors revealing the 3d integrability in the matrix product construction

matrix coordinate bethe ansatz applications to xxz and asep models we present the construction of the full set of eigenvectors of the open asymmetric simple exclusion process asep and xxz models with special constraints on the boundaries the method combines both recent constructions of coordinate bethe ansatz and the old method of matrix ansatz specific to the asep this matrix coordinate bethe ansatz can be viewed as a non commutative coordinate bethe ansatz the non commutative part being related to the algebra appearing in the matrix ansatz

matrix product solution of an inhomogeneous multi species tasep we study a multi species exclusion process with inhomogeneous hopping rates and find a matrix product representation for the stationary state of this model the matrices belong to the tensor algebra of the fundamental quadratic algebra associated with the exclusion process we show that our matrix product representation is equivalent to a graphical construction proposed by ayyer and linusson 2012 arxiv 1206 0316 which generalizes an earlier probabilistic construction due to ferrari and martin 2007 ann prob 35 807

an exact formula for the statistics of the current in the tasep with open boundaries we study the totally asymmetric exclusion process on a finite one dimensional lattice with open boundaries i e in contact with two reservoirs at different potentials the total time integrated current through the system is a random variable that scales linearly with time in the long time limit we conjecture a parametric representation for the generating function of the cumulants of the current which is related to the large deviation function by laplace transform this formula is valid for all system sizes and for all values of the boundary coupling parameters

matrix representation of the stationary measure for the multispecies tasep in this work we construct the stationary measure of the n species totally asymmetric simple exclusion process in a matrix product formulation we make the connection between the matrix product formulation and the queueing theory picture of ferrari and martin in particular in the standard representation the matrices act on the space of queue lengths for n 2 the matrices in fact become tensor products of elements of quadratic algebras this enables us to give a purely algebraic proof of the stationary measure which we present for n 3

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

disordered exclusion process revisited some exact results in the low current regime we study the steady state of the totally asymmetric simple exclusion process with inhomogeneous hopping rates associated with sites site wise disorder using the fact that the non normalized steady state weights which solve the master equation are polynomials in all the hopping rates we propose a general method for calculating their first few lowest coefficients exactly in the case of binary disorder where all slow sites share the same hopping rate r 1 we apply this method to calculate the steady state current up to the quadratic term in r for some particular disorder configurations for the most general non binary disorder we show that in the low current regime the current is determined solely by the current minimizing subset of equal hopping rates provided all other slow rates are much larger our approach can be readily applied to any other driven diffusive system with unidirectional hopping if one can identify a hopping rate such that the current vanishes when this rate is set to zero

exact solution to integrable open multi species ssep and macroscopic fluctuation theory we introduce a multi species generalization of the symmetric simple exclusion process with open boundaries this model possesses the property of being integrable and appears as physically relevant because the boundary conditions can be interpreted as the interaction with particles reservoirs with fixed densities of each species the system is driven out of equilibrium by these reservoirs the steady state is analytically computed in a matrix product form this algebraic structure allows us to obtain exact expressions for the mean particle currents and for the one and two point correlation functions an additivity principle is also derived from the matrix ansatz and permits the computation of the large deviation functional of the density profile we also propose a description of the model in the context of the macroscopic fluctuation theory and we check the consistency with the exact computations from the finite size lattice

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

exact solution of the totally asymmetric simple exclusion process shock profiles the microscopic structure of macroscopic shocks in the one dimensional totally asymmetric simple exclusion process is obtained exactly from the complete solution of the stationary state of a model system containing two types of particles first and second class this nonequilibrium steady state factorizes about any second class particle which implies factorization in the one component system about the random shock position it also exhibits several other interesting features including long range correlations in the limit of zero density of the second class particles the solution also shows that a finite number of second class particles in a uniform background of first class particles form a weakly bound state

exact solution of an exclusion process with three classes of particles and vacancies we present an exact solution for an asymmetric exclusion process on a ring with three classes of particles and vacancies using a matrix product ansatz we find exact expressions for the weights of the configurations in the stationary state the solution involves tensor products of quadratic algebras

invariant measures for a two species asymmetric process we consider a process of two classes of particles jumping on a one dimensional lattice the marginal system of the first class of particles is the one dimensional totally asymmetric simple exclusion process when classes are disregarded the process is also the totally asymmetric simple exclusion process the existence of a unique invariant measure with product marginals with density and for the first and first plus second class particles respectively was shown by ferrari kipnis and saada recently derrida janowsky lebowitz and speer have computed this invariant measure for finite boxes and performed the infinite volume limit based on this computation we give a complete description of the measure and derive some of its properties in particular we show that the invariant measure for the simple exclusion process as seen from a second class particle with asymptotic densities and is equivalent to the product measure with densities to the left of the origin and to the right of the origin

a novel dual field time domain finite element domain decomposition method for computational electromagnetics a novel dual field time domain finite element domain decomposition method is presented for numerical simulation of electromagnetic phenomena the method divides the computation domain into several non overlapping subdomains and computes both the electric and magnetic fields in each subdomain by solving second order vector wave equations a leapfrog like scheme is employed in the time marching to update the alternating electric and magnetic fields adjacent subdomains are related to each other by the equivalent surface currents on the subdomain interfaces which are also updated in a similar fashion the method requires minimum communications between the subdomains thus is highly suitable for parallel computations the proposed domain decomposition method is particularly attractive for the numerical simulation of finite arrays and large scale electromagnetic problems numerical stability and dispersion error analyses are conducted and several radiation examples are presented to demonstrate the accuracy and efficiency of the method

process consistency for adaboost recent experiments and theoretical studies show that adaboost can overfit in the limit of large time if running the algorithm forever is suboptimal a natural question is how low can the prediction error be during the process of adaboost we show under general regularity conditions that during the process of adaboost a consistent prediction is generated which has the prediction error approximating the optimal bayes error as the sample size increases this result suggests that while running the algorithm forever can be suboptimal it is reasonable to expect that some regularization method via truncation of the process may lead to a near optimal performance for sufficiently large sample size

matrix product solution to a 2 species tasep with open integrable boundaries we present an explicit representation for the matrix product ansatz for sometwo species tasep with open boundary conditions the construction relies on theintegrability of the models a property that constrains the possible rates atthe boundaries the realisation is built on a tensor product of copies of thedehp algebras using this explicit construction we are able to calculate thepartition function of the models the densities and currents in the stationarystate are also computed it leads to the phase diagram of the models dependingon the values of the boundary rates we obtain for each species shock waves maximal current or low high densities phases

the matrix product solution of the multispecies partially asymmetric exclusion process we find the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles the solution is in the form of a matrix product representation where the matrices for a system of n species are defined recursively in terms of the matrices for a system of n 1 species a complete proof is given based on the quadratic relations verified by these matrices this matrix product construction is interpreted in terms of the action of a transfer matrix

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

recursive structures in the multispecies tasep we consider a multispecies generalization of the totally asymmetric simple exclusion process tasep with the simple hopping rule for the and th class particles the transition occurs with a rate independent from the values and ferrari and martin 2007 ann prob 35 807 obtained the stationary state of this model thanks to a combinatorial algorithm which was subsequently interpreted as a matrix product representation by evans et al 2009 j stat phys 135 217 this matrix ansatz shows that the stationary state of the multispecies tasep with n classes of particles n tasep can be constructed algebraically by the action of an operator on the n 1 tasep stationary state besides arita et al 2009 j phys a math theor 45 345002 analyzed the spectral structure of the markov matrix they showed that the set of eigenvalues of the n tasep contains those of the n 1 tasep and that the various spectral inclusions can be encoded in a hierarchical set theoretic structure known as the hasse diagram inspired by these works we define nontrivial operators that allow us to construct eigenvectors of the n tasep by lifting the eigenvectors of the n 1 tasep this goal is achieved by generalizing the matrix product representation and the ferrari martin algorithm in particular we show that the matrix ansatz is not only a convenient tool to write the stationary state but in fact intertwines markov matrices of different values of n

dynamical transition in the open boundary totally asymmetric exclusion process we revisit the totally asymmetric simple exclusion process tasep with open boundaries focussing on the recent discovery by de gier and essler that the model has a dynamical transition along a nontrivial line in the phase diagram this line coincides neither with any change in the steady state properties of the tasep nor with the corresponding line predicted by the domain wall theory we provide numerical evidence that the tasep indeed has a dynamical transition along the de gier essler line finding that the most convincing evidence was obtained from the density matrix renormalization group calculations by contrast we find that the dynamical transition is rather difficult to see in direct monte carlo simulations of the tasep we furthermore discuss in general terms scenarios that admit a distinction between the static and dynamic phase behaviour

matrix product formula for macdonald polynomials we derive a matrix product formula for symmetric macdonald polynomials our results are obtained by constructing polynomial solutions of deformed knizhnik zamolodchikov equations which arise by considering representations of the zamolodchikov faddeev and yang baxter algebras in terms of t deformed bosonic operators these solutions are generalised probabilities for particle configurations of the multi species asymmetric exclusion process and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions for weakly increasing compositions anti dominant weights these basis elements coincide with non symmetric macdonald polynomials our formulas imply a natural combinatorial interpretation in terms of solvable lattice models they also imply that normalisations of stationary states of multi species exclusion processes are obtained as macdonald polynomials at q 1

a fast domain decomposition method for solving three dimensional large scale electromagnetic problems an efficient algorithm based on domain decomposition method ddm and partial basic solution vectors pbsv technique is proposed for solving three dimensional 3 d large scale finite periodic electromagnetic problems such as photonic or electromagnetic bandgap structures frequency selective surfaces the entire computational domain is divided into many smaller nonoverlapping subdomains a robin type condition is introduced at the interfaces between subdomains to enforce the field continuity with the help of a set of dual unknowns each subdomain can be tackled independently because of geometric repetitions all the sudomains can be classified into a few building blocks which can be dealt with by an improved pbsv algorithm thus the original problem becomes a much smaller one which involves the unknowns only at the interfaces the resulting linear system of equations is solved by a block symmetric successive over relaxation ssor preconditioned krylov subspace method once the unknowns at the interfaces have been obtained the final solution on each subdomain can easily be calculated independently some numerical examples are provided and show the method is scalable with the number of subdomains

matrix product solution of the stationary state of two species open zero range processes using the matrix product ansatz we obtain solutions of the steady state distribution of the two species open one dimensional zero range process our solution is based on a conventionally employed constraint on the hop rates which eventually allows us to simplify the constituent matrices of the ansatz it is shown that the matrix at each site is given by the tensor product of two sets of matrices and the steady state distribution assumes an inhomogeneous factorized form our method can be generalized to the cases of more than two species of particles

open two species exclusion processes with integrable boundaries we give a complete classification of integrable markovian boundary conditions for the asymmetric simple exclusion process with two species or classes of particles some of these boundary conditions lead to non vanishing particle currents for each species we explain how the stationary state of all these models can be expressed in a matrix product form starting from two key components the zamolodchikov faddeev and ghoshal zamolodchikov relations this statement is illustrated by studying in detail a specific example for which the matrix ansatz involving 9 generators is explicitly constructed and physical observables such as currents densities calculated

exact results for an asymmetric annihilation process with open boundaries we consider a nonequilibrium reaction diffusion model on a finite one dimensional lattice with bulk and boundary dynamics inspired by the glauber dynamics of the ising model we show that the model has a rich algebraic structure that we use to calculate its properties in particular we show that the markov dynamics for a system of a given size can be embedded into the dynamics of systems of higher sizes this remark leads us to devise a technique which we call the transfer matrix ansatz that allows us to determine the steady state distribution and correlation functions furthermore we show that the disorder variables satisfy very simple properties and we give a conjecture for the characteristic polynomial of markov matrices finally we compare the transfer matrix ansatz used here with the matrix product representation of the steady state of one dimensional stochastic models

determinant representation for some transition probabilities in the tasep with second class particles we study the transition probabilities for the totally asymmetric simple exclusion process tasep on the infinite integer lattice with a finite but arbitrary number of first and second class particles using the bethe ansatz we present an explicit expression of these quantities in terms of the bethe wave function in a next step it is proved rigorously that this expression can be written in a compact determinantal form for the case where the order of the first and second class particles does not change in time an independent geometrical approach provides insight into these results and enables us to generalize the determinantal solution to the multi class tasep

on matrix product ground states for reaction diffusion models we discuss a new mechanism leading to a matrix product form for the stationary state of one dimensional stochastic models the corresponding algebra is quadratic and involves four different matrices for the example of a coagulation decoagulation model explicit four dimensional representations are given and exact expressions for various physical quantities are recovered we also find the general structure of n point correlation functions at the phase transition

generalized matrix ansatz in the multispecies exclusion process the partially asymmetric case we investigate one of the simplest multispecies generalizations of the asymmetric simple exclusion process on a ring this process has a rich combinatorial spectral structure and a matrix product form for the stationary state in the totally asymmetric case operators that conjugate the dynamics of systems with different numbers of species were obtained by the authors and recently reported by arita et al 2011 j phys a math theor 44 335004 the existence of such nontrivial operators was reformulated as a representation problem for a specific quadratic algebra generalized matrix ansatz in this work we construct the family of representations explicitly for the partially asymmetric case this solution cannot be obtained by a simple deformation of the totally asymmetric case