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

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

phase transitions in an exactly soluble one dimensional exclusion process we consider an exclusion process with particles injected with rate at the origin and removed with rate at the right boundary of a one dimensional chain of sites the particles are allowed to hop onto unoccupied sites to the right only for the special case of 1 the model was solved previously by derridaet al here we extend the solution to general the phase diagram obtained from our exact solution differs from the one predicted by the mean field approximation

prediction of the mechanical response of the femur with uncertain elastic properties abstract a mandatory requirement for any reliable prediction of the mechanical response of bones based on quantitative computer tomography is an accurate relationship between material properties usually young s modulus e and bone density many such e relationships are available based on different experiments on femur specimens with a large spread due to uncertainties the first goal of this study is to pool and analyze the relevant available experimental data and develop a stochastic e relationship this analysis highlights that there is no experimental data available to cover the entire density range of the human femur and that some popular e relationships are based on data that contains extreme scatter while others are based on a very limited amount of information the second goal is to use the newly developed stochastic e relationship in high order finite element analyses feas for the computation of strains and displacements in two human proximal femurs mimicking in vitro experiments when compared with the experimental observations the fea predictions using the median of the stochastic e relationship follow the underlying distribution of the stochastic e relationship thus most deviations of the fea predictions from experimental observations can possibly be explained by uncertain elastic properties of the femur

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

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

enumeration of totally positive grassmann cells postnikov webs in totally positive grassmann cells in preparation has given a combinatorially explicit cell decomposition of the totally nonnegative part of a grassmannian denoted grk n and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets the main result of our work is an explicit generating function which enumerates the cells in grk n according to their dimension as a corollary we give a new proof that the euler characteristic of grk n is 1 additionally we use our result to produce a new q analog of the eulerian numbers which interpolates between the eulerian numbers the narayana numbers and the binomial coefficients

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

uncertainty quantification for a 1d thermo hyperelastic coupled problem using polynomial chaos projection and p fems numerical solutions of non linear stochastic thermo hyperelastic problems at finite strains are addressed these belong to a category of non linear coupled problems that impose challenges on their numerical treatment both in the physical and stochastic spaces combining the high order finite element methods fems for discretizing the physical space and the polynomial chaos projection pcp method for discretizing the stochastic space a non intrusive scheme is obtained manifesting an exponential convergence rate the method is applied to a 1 d coupled stationary thermo hyperelastic system with stochastic material properties we derive exact stochastic solutions that serve for comparison to numerical results allowing their verification these demonstrate that stochastic coupled problems intractable by standard monte carlo mc methods may be easily computed by combining high order fems with the pcp method controlling discretization errors

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models are discussed

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

covariance kernel representations of multidimensional second order stochastic processes the dynamics of stationary stochastic processes in space is not exactly analogous to that of stationary stochastic processes in the time domain this is due to the unilateral nature of the time series that is only influenced by past values as opposed to the dependence in all directions of the spatial process in this work we unfold the connection that exits between the covariance kernel of a multi dimensional second order autoregressive random process and its underlying discrete random dynamical system starting from a discrete random dynamical system we show that the random process satisfying that system is governed by the modified helmholtz equation in the continuous limit we establish the dependence of the correlation constant on the grid size of the discretization we also show that the random forcing term in the continuous case turns out to be a white noise process a number of covariance functions are worked out for simple and more complex geometrical domains with various boundary conditions in multi dimensions we use both the discrete and the continuous systems in our computations

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

stochastic dynamic analysis of marine risers considering gaussian system uncertainties abstract this paper performs the stochastic dynamic response analysis of marine risers with material uncertainties i e in the mass density and elastic modulus by using stochastic finite element method sfem and model reduction technique these uncertainties are assumed having gaussian distributions the random mass density and elastic modulus are represented by using the karhunen loeve kl expansion the polynomial chaos pc expansion is adopted to represent the vibration response because the covariance of the output is unknown model reduction based on the iterated improved reduced system iirs technique is applied to eliminate the pc coefficients of the slave degrees of freedom to reduce the dimension of the stochastic system monte carlo simulation mcs is conducted to obtain the reference response statistics two numerical examples are studied in this paper the response statistics from the proposed approach are compared with those from mcs it is noted that the computational time is significantly reduced while the accuracy is kept the results demonstrate the efficiency of the proposed approach for stochastic dynamic response analysis of marine risers

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

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

self adjoint operators as functions i lattices galois connections and the spectral order observables of a quantum system described by self adjoint operators in a von neumann algebra or affiliated with it in the unbounded case form a conditionally complete lattice when equipped with the spectral order using this order theoretic structure we develop a new perspective on quantum observables

spectral order of operators and range projections we study the effect algebra i e the positive part of the unit ball of an operator algebra and its relation to the projection lattice from the perspective of the spectral order a spectral orthomorphism is a map between effect algebras which preserves the spectral order and orthogonality of elements we show that if the spectral orthomorphism preserves the multiples of the unit then it is a restriction of a jordan homomorphism between the corresponding algebras this is an optimal extension of the dye s theorem on orthomorphisms of the projection lattices to larger structures containing the projections moreover results on automatic countable additivity of spectral homomorphisms are proved further we study the order properties of the range projection map assigning to each positive contraction in a jbw algebra its range projection it is proved that this map preserves infima of positive contractions in the spectral respectively standard order if and only if the projection lattice of the algebra in question is a modular lattice

a topos foundation for theories of physics i formal languages for physics this paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we discuss two different types of language that can be attached to a system s the first is a propositional language pl s the second is a higher order typed language l s both languages provide deductive systems with an intuitionistic logic the reason for introducing pl s is that as shown in paper ii of the series it is the easiest way of understanding and expanding on the earlier work on topos theory and quantum physics however the main thrust o

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

three dimensional stochastic finite element method for elasto plastic bodies a new stochastic finite element method sfem is formulated for three dimensional softening elasto plastic bodies with random material properties the method is based on the karhunen loeve and polynomial chaos expansions and able to efficiently estimate complete probabilistic characteristics of the response such as moments or pdfs to reduce the computational complexity in the three dimensional setting two alterations are made with respect to the two dimensional sfem proposed earlier by the authors first a variability preserving modification of the karhunen loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties second an efficient algorithm for parallel processing is developed with time consumption being the same order as for an ordinary fem rendering the proposed sfem an effective alternative to monte carlo simulation the applicability of the proposed method to stochastic analysis of strain localization is examined using monte carlo simulation then it is applied to a fault formation problem which is a recent concern of earthquake engineering ground surface layers are modelled by a softening elasto plastic body and the evolution of probabilistic characteristics of the rupture process is analysed in detail some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint copyright 2001 john wiley sons ltd

asymmetric exclusion model and weighted lattice paths we show that the known matrix representations of the stationary state algebra of the asymmetric simple exclusion process asep can be interpreted combinatorially as various weighted lattice paths this interpretation enables us to use the constant term method ctm and bijective combinatorial methods to express many forms of the asep normalization factor in terms of ballot numbers one particular lattice path representation shows that the coefficients in the recurrence relation for the asep correlation functions are also ballot numbers additionally the ctm has a strong combinatorial connection which leads to a new canonical lattice path representation and to the expansion which provides a uniform approach to computing the asymptotic behaviour in the various phases of the asep the path representations enable the asep normalization factor to be seen as the partition function of a more general polymer chain model having a two parameter interaction with a surface we show in the case 1 that the probability of finding a given number of particles in the stationary state can be expressed via non intersecting lattice paths and hence as a simple determinant

nonequilibrium stationary states and equilibrium models with long range interactions it was recently suggested by blythe and evans that a properly defined steady state normalization factor can be seen as a partition function of a fictitious statistical ensemble in which the transition rates of the stochastic process play the role of fugacities in analogy with the lee yang description of phase transition of equilibrium systems they studied the zeros in the complex plane of the normalization factor in order to find phase transitions in nonequilibrium steady states we show that like for equilibrium systems the densities associated with the rates are nondecreasing functions of the rates and therefore one can obtain the location and nature of phase transitions directly from the analytical properties of the densities we illustrate this phenomenon for the asymmetric exclusion process we actually show that its normalization factor coincides with an equilibrium partition function of a walk model in which the densities have a simple physical interpretation

spectral order for unbounded operators abstract the spectral order a notion originated by olson for bounded operators is investigated here in the context of unbounded operators dissimilarities between bounded and unbounded cases are pointed out new criteria for two operators to be comparable are supplied a way of reducing the study of the spectral order to the case of bounded operators is proposed connections with essential selfadjointness are established integral inequalities for monotonically increasing functions are characterized in terms of distribution functions some illustrative examples are furnished

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

formulae for askey wilson moments and enumeration of staircase tableaux we explain how the moments of the weight function of the askey wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors this gives us a direct combinatorial formula for these moments which is related to but more elegant than the formula given in their earlier paper then we use techniques developed by ismail and the third author to give explicit formulae for these moments and for the enumeration of staircase tableaux finally we study the enumeration of staircase tableaux at various specializations of the parameterizations for example we obtain the catalan numbers fibonacci numbers eulerian numbers the number of permutations and the number of matchings

high dimensional integration of smooth functions over cubes we construct a new algorithm for the numerical integration of functions that are defined on a d dimensional cube it is based on the clenshaw curtis rule for d 1 and on smolyak s construction this way we make the best use of the smoothness properties of any nonperiodic function we prove error bounds showing that our algorithm is almost optimal up to logarithmic factors for different classes of functions with bounded mixed derivative numerical results show that the new method is very competitive in particular for smooth integrands and d ge 8

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 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

a topos foundation for theories of physics ii daseinisation and the liberation of quantum theory this paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we study in depth the topos representation of the propositional language pl s for the case of quantum theory in doing so we make a direct link with and clarify the earlier work on applying topos theory to quantum physics the key step is a process we term daseinisation by which a projection operator is mapped to a subobject of the spectral presheaf the topos quantum analog of a classical state space in the second part of the paper we change gear

a topos for algebraic quantum theory the aim of this paper is to relate algebraic quantum mechanics to topos theory so as to construct new foundations for quantum logic and quantum spaces moti vated by bohr s idea that the empirical content of quantum physics is accessible only through classical physics we show how a noncommutative c algebra of observables a induces a topos t a in which the amalgamation of all of its commutative subalgebras comprises a single commutative c algebra a according to the constructive gelfand duality theorem of banaschewski and mulvey the latter has an internal spectrum a in t a which in our approach plays the role of the quantum phase space of the sys tem thus we associate a locale which is the topos theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a heyting algebra to a c algebra which is the noncommutative notion of a space in this setting states on a become probability measures more precisely valuations on and self adjoint elements of a define continuous functions more precisely locale maps fromto scott s interval domain noting that open subsets of a correspond to propositions about the system the pairing map that assigns a generalized truth value to a state and a proposition assumes an extremely simple categorical form formulated in this way the quantum theory defined by a is essentially turned into a classical theory internal to the topos t a these results were inspired by the topos theoretic approach to quantum physics pro posed by butterfield and isham as recently generalized by doring and isham

on the two species asymmetric exclusion process with semi permeable boundaries we investigate the structure of the nonequilibrium stationary state ness of a system of first and second class particles as well as vacancies holes on l sites of a one dimensional lattice in contact with first class particle reservoirs at the boundary sites these particles can enter at site 1 when it is vacant with rate and exit from site l with rate second class particles can neither enter nor leave the system so the boundaries are semi permeable the internal dynamics are described by the usual totally asymmetric exclusion process tasep with second class particles an exact solution of the ness was found by arita here we describe two consequences of the fact that the flux of second class particles is zero first there exist pinned and unpinned fat shocks which determine the general structure of the phase diagram and of the local measures the latter describe the microscopic structure of the system at different macroscopic points in the limit l in terms of superpositions of extremal measures of the infinite system second the distribution of second class particles is given by an equilibrium ensemble in fixed volume or equivalently but more simply by a pressure ensemble in which the pair potential between neighboring particles grows logarithmically with distance we also point out an unexpected feature in the microscopic structure of the ness for finite l if there are n second class particles in the system then the distribution of first class particles respectively holes on the first respectively last n sites is exchangeable

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

multidimensional spectral order for selfadjoint operators abstract the notion of the spectral order is extended to finite families of pairwise commuting bounded and unbounded selfadjoint operators in hilbert space it is shown that the multidimensional spectral order is preserved by transformations represented by spectral integrals of separately increasing borel functions on r in particular the dimensional spectral order is the restriction of product of spectral orders for selfadjoint operators in the context of positive tuples of pairwise commuting selfadjoint operator the relation a b holds if and only if a b for every tuples of nonnegative integer numbers

exact diffusion constant for one dimensional asymmetric exclusion models the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions is considered on a ring of size n the steady state of this system is known all configurations have equal weight which allows for easy computation of the average velocity of a particle in the steady state here an exact expression for the diffusion constant of a particle is obtained for arbitrary number of particles and system size by using a matrix formulation two limits of infinite system size n are discussed firstly when the number of particles remains finite as n to infinity the diffusion constant remains dependent on the exact number of particles due to correlations between successive collisions secondly when the density p of particles is non zero i e when the number of particles is equal to n rho as n to infinity the diffusion constant scales as n 12 the exponent 1 2 is related to the dynamic exponent z 3 2 of the kpz equation in 1 1 dimensions

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

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

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

continued fractions and the partially asymmetric exclusion process we note that a tridiagonal matrix representation of the algebra of the partially asymmetric exclusion process pasep lends itself to interpretation as the transfer matrix for weighted motzkin lattice paths a continued fraction j fraction representation of the lattice path generating function is particularly well suited to discussing the pasep for which the paths have height dependent weights we show that this not only allows a succinct derivation of the normalization and correlation lengths of the pasep but also reveals how finite dimensional representations of the pasep algebra valid only along special lines in the phase diagram relate to the general solution that requires an infinite dimensional representation

numerical solution of spectral stochastic finite element systems abstract this paper addresses the issues involved in solving systems of linear equations which arise in the context of the spectral stochastic finite element ssfem formulation two efficient solution procedures are presented that dramatically reduce the amount of computations involved in numerically solving these problems a brief review is first provided of the underlying spectral approach which highlights the peculiar structure of the matrices generated and how their properties are related to both the level of approximation involved as well as to the convergence behavior of the proposed solution procedure the differences between these matrices from their deterministic finite element counterparts are illustrated an iterative solution scheme is proposed which utilizes their specific properties for efficient memory management and enhanced convergence behavior results from numerical tests are presented comparisons with standard algorithms illustrate the efficiency of the proposed algorithm the second solution procedure presented in this paper is based on hierarchical basis concepts results from numerical tests are again provided and the limitations of this approach are assessed the performance of both proposed algorithms indicates that the linear algebraic systems from the underlying ssfem formulation can be solved with considerably less effort in memory and computation time than their size suggests furthermore the data structures and the hierarchical concept introduced in this study are found to have great potential for the future development of adaptive procedures in stochastic fem

stochastic finite element analysis of shells with combined random material and geometric properties abstract in this work an extended stochastic formulation of the triangular composite facet shell element tric is presented for the case of combined uncertain material young s modulus poisson s ratio and geometric thickness properties these properties are assumed to be described by uncorrelated two dimensional homogeneous stochastic fields the stochastic finite element analysis of shell structures is performed using the spectral representation method for the description of the random fields in conjunction with monte carlo simulation mcs for the computation of the response variability useful conclusions regarding the influence of each one of the structural parameters on the response variability are derived from the numerical tests examined

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

formulation of geometrically non linear problems in the spatial reference frame this paper presents a new formulation for geometrically non linear problems and their numerical treatment by the p version of the finite element method the formulation is characterized by a weak form based on the spatial representation of the equilibrium equations a deformed geometry mapped by the displacement field a natural description for non conservative loads that keeps the symmetry of the equations and a direct non incremental iterative algorithm to solve the equations the paper is concerned only with problems for which the solution is unique i e the load displacement curves are single valued and the strains are small examples are presented 1997 by john wiley sons ltd

a topos foundation for theories of physics iii the representation of physical quantities with arrows this paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system in paper ii we studied the topos representations of the propositional language pl s for the case of quantum theory and in the present paper we do the same thing for the more extensive local language l s one of the main achievements is to find a topos representation for self adjoint operators this involves showing that for any physical quantity a there is an arrow o a r where r is the quantity value object for this theory the construction of o a is an extension of the daseinisation of projection operators that was discussed in paper ii the object r is a monoid object only in the topos setsv h op of the theory and to enhance the applicability of the formalism we apply to r a topos a

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

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

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

finite dimensional representations of the quadratic algebra applications to the exclusion process we study the one dimensional partially asymmetric simple exclusion process asep with open boundaries that describes a system of hard core particles hopping stochastically on a chain coupled to reservoirs at both ends derrida and coworkers showed in 1993 that the stationary probability distribution of this model can be represented as a trace on a quadratic algebra closely related to the deformed oscillator algebra we construct all finite dimensional irreducible representations of this algebra this enables us to compute the stationary bulk density as well as all correlation lengths for the asep on a set of special curves of the phase diagram

an exact solution of a one dimensional asymmetric exclusion model with open boundaries a simple asymmetric exclusion model with open boundaries is solved exactly in one dimension the exact solution is obtained by deriving a recursion relation for the steady state if the steady state is known for all system sizes less thann then our equation 8 gives the steady state for sizen using this recursion we obtain closed expressions 48 for the average occupations of all sites the results are compared to the predictions of a mean field theory in particular for infinitely large systems the effect of the boundary decays as the distance to the power 1 2 instead of the inverse of the distance as predicted by the mean field theory

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

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

return polynomials for non intersecting paths above a surface on the directed square lattice we enumerate sets of n non intersecting t step paths on the directed square lattice which are excluded from the region below the surface y 0t o which they are initially attached in particular we obtain a product formula for the number of star configurations in which the paths have arbitrary fixed endpoints we also consider the return polynomial r w y m0 r w y m m

stochastic elastic plastic finite elements abstract a computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic plastic materials uncertain elastic plastic material properties are modeled as random fields which appear as the coefficient term in the governing partial differential equation of mechanics a spectral stochastic elastic plastic finite element method with fokker planck kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non linear elastic plastic partial differential equation with stochastic coefficient to this end the random field material properties are discretized in both spatial and stochastic dimension into finite numbers of independent basic random variables using karhunen loeve expansion those random variables are then propagated through the elastic plastic constitutive rate equation using fokker planck kolmogorov equation approach to obtain the evolutionary material properties as the material plastifies the unknown displacement solution random field is then assembled using polynomial chaos as a function of known basic random variables and unknown deterministic coefficients which are obtained by minimizing the error of finite representation by galerkin technique

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

topos models for physics and topos theory what is the role of topos theory in the topos models for quantum theory as used by isham butterfield doring heunen landsman spitters and others in other words what is the interplay between physical motivation for the models and the mathematical framework used in these models concretely we show that the presheaf topos model of butterfield isham and doring resembles classical physics when viewed from the internal language of the presheaf topos similar to the copresheaf topos model of heunen landsman and spitters both the presheaf and copresheaf models provide a quantum logic in the form of a complete heyting algebra although these algebras are natural from a topos theoretic stance we seek a physical interpretation for the logical operations finally we investigate dynamics in particular we describe how an automorphism on the operator algebra induces a homeomorphism or isomorphism of locales on the associated state spaces of the topos models and how elementary propositions and truth values transform under the action of this homeomorphism also with dynamics the focus is on the internal perspective of the topos

stochastic simulation of riser sections with uncertain measured pressure loads and or uncertain material properties abstract we investigate three dimensional problems in solid mechanics with stochastic loading or material properties to solve these problems we use a spectral expansion of the solution and random inputs based on askey type orthogonal polynomials in terms of independent identically distributed i i d random variables a galerkin procedure using these types of expansions the generalized polynomial chaos gpc method is employed to solve linear elasticity problems an analagous spectral collocation formulation is used to study problems in nonlinear elasticity these methods both cast the stochastic problem as a coupled or decoupled high dimensional system of deterministic pdes which is then solved numerically using a deterministic p finite element solver we present algorithms for solving certain coupled systems arising from the stochastic galerkin projection without modifying the original deterministic solver three dimensional riser sections undergoing elastic deformations due to random pressure loads are considered we also model a riser section with stochastic young s modulus undergoing deterministic loads it is demonstrated that the gpc method provides accurate and efficient results at a speed up factor of two and three orders of magnitude compared to traditional monte carlo simulations for nonlinear problems the stochastic collocation method is also shown to be much faster than monte carlo simulation while still rivaling this method in simplicity of implementation

current fluctuations in the one dimensional symmetric exclusion process with open boundaries we calculate the first four cumulants of the integrated current of the one dimensional symmetric simple exclusion process of n sites with open boundary conditions for large system size n the generating function of the integrated current depends on the densities a and b of the two reservoirs and on the fugacity z the parameter conjugated to the integrated current through a single parameter based on our expressions for these first four cumulants we make a conjecture which leads to a prediction for all the higher cumulants in the case a 1 and b 0 our conjecture gives the same universal distribution as the one obtained by lee levitov and yakovets for one dimensional quantum conductors in the metallic regime

large deviation of the density profile in the steady state of the open symmetric simple exclusion process we consider an open one dimensional lattice gas on sites i 1 n with particles jumping independently with rate 1 to neighboring interior empty sites the simple symmetric exclusion process the particle fluxes at the left and right boundaries corresponding to exchanges with reservoirs at different chemical potentials create a stationary nonequilibrium state sns with a steady flux of particles through the system the mean density profile in this state which is linear describes the typical behavior of a macroscopic system i e this profile occurs with probability 1 when n the probability of microscopic configurations corresponding to some other profile x x i n has the asymptotic form exp n f f is the large deviation functional in contrast to equilibrium systems for which f eq is just the integral of the appropriately normalized local free energy density the f we find here for the nonequilibrium system is a nonlocal function of this gives rise to the long range correlations in the sns predicted by fluctuating hydrodynamics and suggests similar non local behavior of f in general sns where the long range correlations have been observed experimentally