asymptotics in asep with step initial condition in previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition particles beginning at the positive integers there it was shown that the probability distribution for the position of an individual particle is given by an integral whose integrand involves a fredholm determinant here we use this formula to obtain three asymptotic results for the positions of these particles in one an apparently new distribution function arises and in another the distribution function f2 arises the latter extends a result of johansson on tasep to asep and hence proves kpz universality for asep with step initial condition

three dimensional non planar crack growth by a coupled extended finite element and fast marching method a numerical technique for non planar three dimensional linear elastic crack growth simulations is proposed this technique couples the extended finite element method x fem and the fast marching method fmm in crack modeling using x fem the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two dimensional asymptotic crack tip displacement fields the initial crack geometry is represented by two level set functions and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation crack modeling is performed without the need to mesh the crack and crack propagation is simulated without remeshing crack growth is conducted using fmm unlike a level set formulation for interface capturing no iterations nor any time step restrictions are imposed in the fmm planar and non planar quasi static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique copyright 2008 john wiley sons ltd

a parallel two scale method for eikonal equations numerous applications of eikonal equations prompted the development of many efficient numerical algorithms the heap cell method hcm is a recent serial two scale technique that has been shown to have advantages over other serial state of the art solvers for a wide range of problems a chacon and a vladimirsky siam j sci comput 34 2012 pp a547 a578 this paper presents a parallelization of hcm for a shared memory architecture the numerical experiments in r 3 show that the parallel hcm exhibits good algorithmic behavior and scales well resulting in a very fast and practical solver

noncolliding brownian motion with drift and time dependent stieltjes wigert determinantal point process using the determinantal formula of biane bougerol and o connell we give multitime joint probability densities to the noncolliding brownian motion with drift where the number of particles is finite we study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well ordered with a scale we show that at each time t 0 the single time probability density of particle system is exactly transformed to the biorthogonal stieltjes wigert matrix model in the chern simons theory introduced by dolivet and tierz here one parameter extensions extensions of the stieltjes wigert polynomials which are themselves q extensions of the hermite polynomials play an essential role the two parameters a and of the process combined with time t are mapped to the parameters q and of the biorthogonal polynomials by the transformation of normalization factor of our probability density the partition function of the chern simons matrix model is readily obtained we study the determinantal structure of the matrix model and prove that at each time t 0 the present noncolliding brownian motion with drift is a determinantal point process in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel using the obtained correlation kernel we study time evolution of the noncolliding brownian motion with drift

an effective algorithm for the generation of patient specific purkinje networks in computational electrocardiology the purkinje network is responsible for the fast and coordinated distribution of the electrical impulse in the ventricle that triggers its contraction therefore it is necessary to model its presence to obtain an accurate patient specific model of the ventricular electrical activation in this paper we present an efficient algorithm for the generation of a patient specific purkinje network driven by measures of the electrical activation acquired on the endocardium the proposed method provides a correction of an initial network generated by means of a fractal law and it is based on the solution of eikonal problems both in the muscle and in the purkinje network we present several numerical results both in an ideal geometry with synthetic data and in a real geometry with patient specific clinical measures these results highlight an improvement of the accuracy provided by the patient specific purkinje network with respect to the initial one in particular a cross validation test shows an accuracy increase of 19 when only the 3 of the total points are used to generate the network whereas an increment of 44 is observed when a random noise equal to 20 of the maximum value of the clinical data is added to the measures

baxter operator and archimedean hecke algebra in this paper we introduce baxter integral mathcal q operators for finite dimensional lie algebras mathfrak gl ell 1 and mathfrak so 2 ell 1 whittaker functions corresponding to these algebras are eigenfunctions of the mathcal q operators with the eigenvalues expressed in terms of gamma functions the appearance of the gamma functions is one of the manifestations of an interesting connection between mellin barnes and givental integral representations of whittaker functions which are in a sense dual to each other we define a dual baxter operator and derive a family of mixed mellin barnes givental integral representations givental and mellin barnes integral representations are used to provide a short proof of the friedberg bump and bump conjectures for g gl l 1 proved earlier by stade we also identify eigenvalues of the baxter mathcal q operator acting on whittaker functions with local archimedean l factors the baxter mathcal q operator introduced in this paper is then described as a particular realization of the explicitly defined universal baxter operator in the spherical hecke algebra mathcal h g mathbb r k k being a maximal compact subgroup of g finally we stress an analogy between mathcal q operators and certain elements of the non archimedean hecke algebra mathcal h g mathbb q p g mathbb z p

tropical combinatorics and whittaker functions we establish a fundamental connection between the geometric robinson schensted knuth rsk correspondence and gl n r whittaker functions analogous to the well known relationship between the rsk correspondence and schur functions this gives rise to a natural family of measures associated with gl n r whittaker functions which are the analogues in this setting of the schur measures on integer partitions the corresponding analogue of the cauchy littlewood identity can be seen as a generalization of an integral identity for gl n r whittaker functions due to bump and stade as an application we obtain an explicit integral formula for the laplace transform of the law of the partition function associated with a 1 dimensional directed polymer model with log gamma weights recently introduced by one of the authors

unstable recurrent patterns in kuramoto sivashinsky dynamics we undertake an exploration of recurrent patterns in the antisymmetric subspace of the one dimensional kuramoto sivashinsky system for a small but already rather turbulent system the long time dynamics takes place on a low dimensional invariant manifold a set of equilibria offers a coarse geometrical partition of this manifold the newton descent method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions the attracting set appears surprisingly thin its backbone consists of several smale horseshoe repellers well approximated by intrinsic local one dimensional return maps each with an approximate symbolic dynamics the dynamics appears decomposable into chaotic dynamics within such local repellers interspersed by rapid jumps between them

geometric rsk correspondence whittaker functions and symmetrized random polymers we show that the geometric lifting of the rsk correspondence introduced by a n kirillov physics and combinatorics proc nagoya 2000 2nd internat workshop pp 82 150 2001 is volume preserving with respect to a natural product measure on its domain and that the integrand in givental s integral formula for mathit gl n mathbb r whittaker functions arises naturally in this context apart from providing further evidence that whittaker functions are the natural analogue of schur polynomials in this setting our results also provide a new combinatorial framework for the study of random polymers when the input matrix consists of random inverse gamma distributed weights the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of whittaker functions next we restrict the geometric rsk mapping to symmetric matrices and show that the volume preserving property continues to hold we determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal with an additional factor on the main diagonal the third combinatorial mapping studied is a variant of the geometric rsk mapping for triangular arrays which is again showed to be volume preserving this leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal we also show that the analogues of the cauchy littlewood identity in the setting of this paper are equivalent to a collection of whittaker integral identities conjectured by bump number theory trace formulas and discrete groups pp 49 109 1989 and bump and friedberg festschrift in honor of piatetski shapiro part ii pp 47 65 1990 and proved by stade am j math 123 121 161 2001 israel j math 127 201 219 2002 our approach leads to new combinatorial proofs and generalizations of these identities with some restrictions on the parameters

mellin transforms of gl n r whittaker functions using a known recursive formula for the class one principal series gl n r whittaker function we deduce a recursive formula for the multiple mellin transform of this function from the latter formula we verify a conjecture of goldfeld regarding the location of poles of our mellin transform we further express the residues at these poles in terms of mellin transforms of lower rank whittaker functions our next result concerns the simplification of our mellin transform under a certain restriction on the transform parameter we show by applying a change of variable to our above result on poles of the mellin transform that the gl n r transform reduces essentially to a gl n 1 r transform under this restriction we then demonstrate that under further restriction of the mellin transform parameter this mellin transform in fact reduces to a ratio of products of gamma functions our result proves a conjecture of bump and friedberg that is motivated by the theory of exterior square automorphic l functions finally we show that a certain mellin transform of a product of two whittaker functions one on gl n 1 r and the other on gl n r reduces to a product of gamma functions this last result verifies a conjecture of bump regarding archimedean euler factors of automorphic l functions on gl n 1 r gl n r

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

a gradient augmented level set method for unstructured grids we present and test a gradient augmented level set method for unstructured grids which has been designed to allow for easy integration in parallel flow solvers the means to this end is a hermite formulation where the gradient is stored and advected as an independent variable along with the level set function the method uses narrow band storage in conjunction with a fast marching method which is also designed to take advantage of the gradient augmentation we demonstrate through standard passive advection test cases that the proposed method is able to compute interface motion with performance comparable to popular high performing methods on structured grids

fluctuations for stationary q tasep we consider the q totally asymmetric simple exclusion process q tasep in the stationary regime and study the fluctuation of the position of a particle we first observe that the problem can be studied as a limiting case of an n particle q tasep with a random initial condition and with particle dependent hopping rate then we explain how this n particle q tasep can be encoded in a dynamics on a two sided gelfand tsetlin cone described by a two sided q whittaker process and present a fredholm determinant formula for the q laplace transform of the position of a particle two main ingredients in its derivation is the ramanujan s bilateral summation formula and the cauchy determinant identity for the theta function with an extra parameter based on this we establish that the position of a particle obeys the universal stationary kpz distribution the baik rains distribution in the long time limit

log gamma polymer free energy fluctuations via a fredholm determinant identity we prove that under n1 3 scaling the limiting distribution as n of the free energy of seppalainen s log gamma discrete directed polymer is gue tracy widom the main technical innovation we provide is a general identity between a class of n fold contour integrals and a class of fredholm determinants applying this identity to the integral formula proved in corwin et al tropical combinatorics and whittaker functions http arxiv org abs 1110 3489v3 math pr 2012 for the laplace transform of the log gamma polymer partition function we arrive at a fredholm determinant which lends itself to asymptotic analysis and thus yields the free energy limit theorem the fredholm determinant was anticipated in borodin and corwin macdonald processes http arxiv org abs 1111 4408v3 math pr 2012 via the formalism of macdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach

determinantal processes with number variance saturation consider dyson s hermitian brownian motion model after a finite time s where the process is started at n equidistant points on the real line these n points after time s form a determinantal process and has a limit as n this limting determinantal process has the interesting feature that it shows number variance saturation the variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity number variance saturation is also seen for example in the zeros of the riemann function 21 2 the process can also be constructed using non intersecting paths and we consider several variants of this construction one construction leads to a model which shows a transition from a non universal behaviour with number variance saturation to a universal sine kernel behaviour as we go up the line

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

determinantal structures in the o connell yor directed random polymer model we study the semi discrete directed random polymer model introduced by o connell and yor we obtain a representation for the moment generating function of the polymer partition function in terms of a determinantal measure this measure is an extension of the probability measure of the eigenvalues for the gaussian unitary ensemble in random matrix theory to establish the relation we introduce another determinantal measure on larger degrees of freedom and consider its few properties from which the representation above follows immediately

stochastic higher spin six vertex model and q taseps abstract we present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the macdonald processes hierarchy first we show how macdonald q difference operators with t 0 an algebraic tool crucial for studying the corresponding macdonald processes can be utilized to get q moments of the height function h in the higher spin six vertex model first computed in 21 using bethe ansatz this result in particular implies that for the vertex model with the step bernoulli boundary condition the value of h at an arbitrary point n 1 t z 2 z 1 has the same distribution as the last component n of a random partition under a specific t 0 macdonald measure on the other hand it is known that x n n n can be identified with the location of the nth particle in a certain discrete time q tasep started from the step initial configuration the second construction we present is a coupling of this q tasep and the higher spin six vertex model with the step bernoulli boundary condition along time like paths providing an independent probabilistic explanation of the equality of h n 1 t and x n n in distribution as an illustration of our main results we obtain gue tracy widom asymptotics of a certain discrete time q tasep with the step initial configuration and special jump parameters by means of schur measures which are t q macdonald measures this analysis combines our results with the identification of averages of observables between the stochastic higher spin six vertex model and schur measures obtained recently in 8

integral representations for the eigenfunctions of quantum open and periodic toda chains from the qism formalism the integral representations for the eigenfunctions of n particle quantum open and periodic toda chains are constructed within the framework of the quantum inverse scattering method both periodic and open n particle solutions have essentially the same structure being written as a generalized fourier transform over the eigenfunctions of the n 1 particle open toda chain with the kernels satisfying the baxter equations of second and first order respectively in the latter case this leads to recurrent relations which result in a representation of mellin burnes type solutions of an open chain as a byproduct we obtain the gindikin karpelevich formula for the harish chandra function in the case of the gl n group

a new incompressibility discretization for a hybrid particle mac grid representation with surface tension we take a particle based approach to incompressible free surface flow motivated by the fact that an explicit representation of the interface geometry and internal deformations gives precise feedback to an implicit solver for surface tension methods that enforce incompressibility directly on the particles are typically numerically inefficient compared to those that utilize a background grid however background grid discretizations suffer from inaccuracy near the free surface where they do not properly capture the interface geometry therefore our incompressibility discretization utilizes a particle based projection near the interface and a background mac grid based projection for efficiency in the vast interior of the liquid domain as well as a novel method for coupling these two disparate projections together we show that the overall coupled elliptic solver is second order accurate and remains second order accurate when used in conjunction with an appropriate temporal discretization for parabolic problems a similar second order accurate discretization is derived when the mac grid unknowns are located on faces as opposed to cell centers so that navier stokes viscosity can be solved for implicitly as well finally we present a fully implicit approach to surface tension that is robust enough to achieve a steady state solution in a single time step beyond stable implicit surface tension for our novel hybrid discretization we demonstrate preliminary results for both standard front tracking and the particle level set method

a path transformation for random walks and the robinson schensted correspondence the author and marc yor recently introduced a path transformation g k with the property that for x belonging to a certain class of random walks on z k the transformed walk g k x has the same law as the original walk conditioned never to exit the weyl chamber x x 1 less than or equal to less than or equal to x k in this paper we show that g k is closely related to the robinson schensted algorithm and use this connection to give a new proof of the above representation theorem the new proof is valid for a larger class of random walks and yields additional information about the joint law of x and g k x the corresponding results for the brownian model are recovered by donsker s theorem these are connected with hermitian brownian motion and the gaussian unitary ensemble of random matrix theory the connection we make between the path transformation g k and the robinson schensted algorithm also provides a new formula and interpretation for the latter this can be used to study properties of the robinson schensted algorithm and moreover extends easily to a continuous setting

gues and queues consider the process d k k 1 2 given by b i being independent standard brownian motions this process describes the limiting behavior near the edge in queues in series totally asymmetric exclusion processes or oriented percolation the problem of finding the distribution of d was posed in gw the main result of this paper is that the process d has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from gaussian unitary ensemble

the q pushasep a new integrable model for traffic in 1 1 dimension we introduce a new interacting stochastic particle system q pushasep which interpolates between the q tasep of borodin and corwin probab theory relat fields 158 1 2 225 400 2014 see also borodin et al ann probab 42 6 2314 2382 2014 borodin and corwin int math res not 2 499 537 2015 o connell and pei electron j probab 18 95 1 25 2013 borodin et al comput math 2013 and the q pushtasep introduced recently borodin and petrov adv math 2013 in the q pushasep particles can jump to the left or to the right and there is a certain partially asymmetric pushing mechanism present this particle system has a nice interpretation as a model of traffic on a one lane highway using the quantum many body system approach we explicitly compute the expectations of a large family of observables for this system in terms of nested contour integrals we also discuss relevant fredholm determinantal formulas for the distribution of the location of each particle and connections of the model with a certain two sided version of macdonald processes and with the semi discrete stochastic heat equation

the asep and determinantal point processes we introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line with correlation kernels defined via spectral projections for the associated jacobi matrices for classical weights we show how such ensembles arise as limits of various hypergeometric orthogonal polynomial ensembles we then prove that the q laplace transform of the height function of the asep with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete laguerre ensemble a member of the new family this allows us to obtain the large time asymptotics of the asep in three limit regimes a for finitely many rightmost particles b gue tracy widom asymptotics of the height function c kpz asymptotics of the height function for the asep with weak asymmetry we also give similar results for two instances of the stochastic six vertex model in a quadrant the proofs are based on limit transitions for the corresponding determinantal point processes

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

solving the eikonal equation on an adaptive mesh a fast marching method fmm using the line of site calculation to solve the eikonal equation is applied to an adaptive mesh the criteria for refinement are the curvature of the propagating front it is shown empirically that for cases involving an initial front initiated from a single point in an open three dimensional domain and constant front propagating speed that the fmm with adaptive mesh refinement amr uses roughly an order of magnitude less cpu time and an order of magnitude less cpu memory than the non amr fmm to attain a similar level of accuracy it is also shown that the amr fmm refines when the curvature is caused by boundary irregularities and also non constant front propagating speed

spectral theory for interacting particle systems solvable by coordinate bethe ansatz we develop spectral theory for the q hahn stochastic particle system introduced recently by povolotsky that is we establish a plancherel type isomorphism result that implies completeness and biorthogonality statements for the bethe ansatz eigenfunctions of the system owing to a markov duality with the q hahn tasep a discrete time generalization of tasep with particles jump distribution being the orthogonality weight for the classical q hahn orthogonal polynomials we write down moment formulas that characterize the fixed time distribution of the q hahn tasep with general initial data the bethe ansatz eigenfunctions of the q hahn system degenerate into eigenfunctions of other not necessarily stochastic interacting particle systems solvable by the coordinate bethe ansatz this includes the asep the asymmetric six vertex model and the heisenberg xxz spin chain all models are on the infinite lattice in this way each of the latter systems possesses a spectral theory too in particular biorthogonality of the asep eigenfunctions which follows from the corresponding q hahn statement implies symmetrization identities of tracy and widom for asep with either step or step bernoulli initial configuration as corollaries another degeneration takes the q hahn system to the q boson particle system dual to q tasep studied in detail in our previous paper 2013 thus at the spectral theory level we unify two discrete space regularizations of the kardar parisi zhang equation stochastic heat equation namely q tasep and asep

identification of contact regions in semiconductor transistors by level set methods in this paper we present the formulation of level set methods for the inverse problem of identifying an interface in the coefficient of an elliptic equation from a boundary measurement this problem arises from the modeling of the identification of contact regions by boundary measurements for semiconductor transistors we propose the gauss newton direction as the interface velocity and implement the scheme for a parameterized class of interfaces

brownian gibbs property for airy line ensembles consider a collection of n brownian bridges b i n n to mathbb r b i n b i n 0 1 i n conditioned not to intersect the edge scaling limit of this system is obtained by taking a weak limit as n of the collection of curves scaled so that the point 0 21 2 n is fixed and space is squeezed horizontally by a factor of n 2 3 and vertically by n 1 3 if a parabola is added to each of the curves of this scaling limit an x translation invariant process sometimes called the multi line airy process is obtained we prove the existence of a version of this process which we call the airy line ensemble in which the curves are almost surely everywhere continuous and non intersecting this process naturally arises in the study of growth processes and random matrix ensembles as do related processes with wanderers and outliers we formulate our results to treat these relatives as well note that the law of the finite collection of brownian bridges above has the property called the brownian gibbs property of being invariant under the following action select an index 1 k n and erase b k on a fixed time interval a b n n then replace this erased curve with a new curve on a b according to the law of a brownian bridge between the two existing endpoints a b k a and b b k b conditioned to intersect neither the curve above nor the one below we show that this property is preserved under the edge scaling limit and thus establish that the airy line ensemble has the brownian gibbs property an immediate consequence of the brownian gibbs property is a confirmation of the prediction of m prahofer and h spohn that each line of the airy line ensemble is locally absolutely continuous with respect to brownian motion we also obtain a proof of the long standing conjecture of k johansson that the top line of the airy line ensemble minus a parabola attains its maximum at a unique point this establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi line airy process and readily yields several other interesting properties of this process

local or global minima flexible dual front active contours most variational active contour models are designed to find local minima of data dependent energy functionals with the hope that reasonable initial placement of the active contour will drive it toward a desirable local minimum as opposed to an undesirable configuration due to noise or complex image structure as such there has been much research into the design of complex region based energy functionals that are less likely to yield undesirable local minima when compared to simpler edge based energy functionals whose sensitivity to noise and texture is significantly worse unfortunately most of these more robust region based energy functionals are applicable to a much narrower class of imagery compared to typical edge based energies due to stronger global assumptions about the underlying image data devising new implementation algorithms for active contours that attempt to capture more global minimizers of already proposed image based energies would allow us to choose an energy that makes sense for a particular class of energy without concern over its sensitivity to local minima such implementations have been proposed for capturing global minima however sometimes the completely global minimum is just as undesirable as a minimum that is too local in this paper we propose a novel fast and flexible dual front implementation of active contours motivated by minimal path techniques and utilizing fast sweeping algorithms which is easily manipulated to yield minima with variable degrees of localness and globalness by simply adjusting the size of active regions the ability to gracefully move from capturing minima that are more local according to the initial placement of the active contour surface to minima that are more global allows this model to more easily obtain desirable minimizers which often are neither the most local nor the most global experiments on various 2d and 3d images and comparisons with some active contour models and region growing methods are also given to illustrate the properties of this model and its performance in a variety of segmentation applications

system of complex brownian motions associated with the o connell process the o connell process is a softened version a geometric lifting with a parameter a 0 of the noncolliding brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length a this process is not determinantal under a special entrance law however borodin and corwin gave a fredholm determinant expression for the expectation of an observable which is a softening of an indicator of a particle position we rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and show if the number of particles is n the rank of the matrix of the fredholm determinant is n then we give a representation for the quantity by using an n particle system of complex brownian motions cbms the complex function which gives the determinantal expression to the weight of cbm paths is not entire but in the combinatorial limit a 0 it becomes an entire function providing conformal martingales and the cbm representation for the noncolliding brownian motion is recovered

new formulas for whittaker functions on gl n r abstract we derive new recursive formulas for principal series whittaker functions of both fundamental and class one type on gl n r these formulas relate such whittaker functions to their counterparts on gl n 1 r instead of on gl n 2 r as has been the case in numerous earlier studies in the particular case n 3 and for certain special values of the eigenvalues our formulas are seen to resemble classical summation formulas due to gegenbauer for bessel functions to illuminate this resemblance we also derive in this work formulas for certain special values of gl 3 r fundamental whittaker functions these formulas may be understood as analogs of expressions derived by bump and friedberg for class one whittaker functions on gl 3 r

transversal fluctuations of the asep stochastic six vertex model and hall littlewood gibbsian line ensembles we consider the asep and the stochastic six vertex model started with step initial data after a long time t it is known that the one point height function fluctuations for these systems are of order t1 3 we prove the kpz prediction of t2 3 scaling in space namely we prove tightness and brownian absolute continuity of all subsequential limits as t goes to infinity of the height function with spatial coordinate scaled by t2 3 and fluctuations scaled by t1 3 the starting point for proving these results is a connection discovered recently by borodin bufetov wheeler between the stochastic six vertex height function and the hall littlewood process a certain measure on plane partitions interpreting this process as a line ensemble with a gibbsian resampling invariance we show that the one point tightness of the top curve can be propagated to the tightness of the entire curve

paths in weyl chambers and random matrices baryshnikov 4 and gravner tracy widom 15 have shown that the largest eigenvalue of a random matrix of the g u e of order d has the same distribution as

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

optimal control with budget constraints and resets we consider both discrete and continuous control problems constrained by a fixed budget of some resource which may be renewed upon entering a preferred subset of the state space in the discrete case we consider deterministic shortest path problems on graphs with a full budget reset in all preferred nodes in the continuous case we derive augmented pdes of optimal control which are then solved numerically on the extended state space with a full instantaneous budget reset on the preferred subset we introduce an iterative algorithm for solving these problems efficiently the method s performance is demonstrated on a range of computational examples including optimal path planning with constraints on prolonged visibility by a static enemy observer

robust second order accurate discretizations of the multi dimensional heaviside and dirac delta functions we present a robust second order accurate method for discretizing the multi dimensional heaviside and the dirac delta functions on irregular domains the method is robust in the following ways 1 integrations of source terms on a co dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain s boundary 2 the method depends only on the function value of a level function not on its derivatives we present the discretizations in tabulated form to make their implementations straightforward we present numerical results in two and three spatial dimensions to demonstrate the second order accuracy in the l 1 norm in the case of the solution of pdes with singular source terms in the case of evaluating the contribution of singular source terms on interfaces the method is also second order accurate in the l norm

a multi layer extension of the stochastic heat equation motivated by recent developments on solvable directed polymer models we define a multi layer extension of the stochastic heat equation involving non intersecting brownian motions by developing a connection with darboux transformations and the two dimensional toda equations we conjecture a markovian evolution in time for this multi layer process as a first step in this direction we establish an analogue of the karlin mcgregor formula for the stochastic heat equation and use it to prove a special case of this conjecture

stabilization of parametric active contours using a tangential redistribution term depending on implementation active contours have been classified as geometric or parametric active contours parametric contours irrespective of representation are known to suffer from the problem of irregular bunching and spacing out of curve points during the curve evolution in a spline based implementation of active contours this leads to occasional formation of loops locally and subsequently the curve blows up due to instabilities in this paper we analyze the reason for this problem and propose a solution to alleviate the same we propose an ordinary differential equation ode for controlling the curve parametrization during evolution by including a tangential force we show that the solution of the proposed ode is bounded we demonstrate the effectiveness of the proposed method for segmentation and tracking tasks on closed as well as open contours

directed polymers and the quantum toda lattice we characterize the law of the partition function of a brownian directed polymer model in terms of a diffusion process associated with the quantum toda lattice the proof is via a multidimensional generalization of a theorem of matsumoto and yor concerning exponential functionals of brownian motion it is based on a mapping which can be regarded as a geometric variant of the rsk correspondence

from duality to determinants for q tasep and asep we prove duality relations for two interacting particle systems the q deformed totally asymmetric simple exclusion process q tasep and the asymmetric simple exclusion process asep expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents respectively duality implies that they solve systems of odes these systems are integrable and for particular step and half stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems solutions and hence also the moments we form laplace transform like generating functions of these moments and via residue calculus we compute two different types of fredholm determinant formulas for such generating functions for asep the first type of formula is new and readily lends itself to asymptotic analysis as necessary to reprove gue tracy widom distribution fluctuations for asep while the second type of formula is recognizable as closely related to tracy and widom s asep formula comm math phys 279 2008 815 844 j stat phys 132 2008 291 300 comm math phys 290 2009 129 154 j stat phys 140 2010 619 634 for q tasep both formulas coincide with those computed via borodin and corwin s macdonald processes probab theory related fields 2014 158 225 400 both q tasep and asep have limit transitions to the free energy of the continuum directed polymer the logarithm of the solution of the stochastic heat equation or the hopf cole solution to the kardar parisi zhang equation thus q tasep and asep are integrable discretizations of these continuum objects the systems of odes associated to their dualities are deformed discrete quantum delta bose gases and the procedure through which we pass from expectations of their duality functionals to characterizing generating functions is a rigorous version of the replica trick in physics

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

point to line polymers and orthogonal whittaker functions we study a one dimensional directed polymer model in an inverse gamma random environment known as the log gamma polymer in three different geometries point to line point to half line and when the polymer is restricted to a half space with end point lying free on the corresponding half line via the use of a n kirillov s geometric robinson schensted knuth correspondence we compute the laplace transform of the partition functions in the above geometries in terms of orthogonal whittaker functions thus obtaining new connections between the ubiquitous class of whittaker functions and exactly solvable probabilistic models in the case of the first two geometries we also provide multiple contour integral formulae for the corresponding laplace transforms passing to the zero temperature limit we obtain new formulae for the corresponding last passage percolation problems with exponential weights

a fredholm determinant representation in asep in previous work tracy and widom in commun math phys 279 815 844 2008 the authors found integral formulas for probabilities in the asymmetric simple exclusion process asep on the integer lattice the dynamics are uniquely determined once the initial state is specified in this note we restrict our attention to the case of step initial condition with particles at the positive integers and consider the distribution function for the mth particle from the left in tracy and widom commun math phys 279 815 844 2008 an infinite series of multiple integrals was derived for the distribution in this note we show that the series can be summed to give a single integral whose integrand involves a fredholm determinant we use this determinant representation to derive non rigorously at this writing a scaling limit

anisotropic 2 1 d growth and gaussian limits of q whittaker processes we consider a discrete model for anisotropic 2 1 dimensional growth of an interface height function owing to a connection with q whittaker functions this system enjoys many explicit integral formulas by considering certain gaussian stochastic differential equation limits of the model we are able to prove a space time limit of covariances to those of the 2 1 dimensional additive stochastic heat equation or edwards wilkinson equation along characteristic directions in particular the bulk height function converges to the gaussian free field which evolves according to this stochastic pde

stochastic higher spin vertex models on the line we introduce a four parameter family of interacting particle systems on the line which can be diagonalized explicitly via a complete set of bethe ansatz eigenfunctions and which enjoy certain markov dualities using this for the systems started in step initial data we write down nested contour integral formulas for moments and fredholm determinant formulas for laplace type transforms taking various choices or limits of parameters this family degenerates to many of the known exactly solvable models in the kardar parisi zhang universality class as well as leads to many new examples of such models in particular asymmetric simple exclusion process the stochastic six vertex model q totally asymmetric simple exclusion process and various directed polymer models all arise in this manner our systems are constructed from stochastic versions of the r matrix related to the six vertex model one of the key tools used here is the fusion of r matrices and we provide a probabilistic proof of this procedure

estimation of reactive fluxes in gradient stochastic systems using an analogy with electric circuits abstract we propose an approach for finding dominant reactive channels and calculating percentages of reactive flux through each channel in chemical systems driven by a deterministic potential force and a small thermal noise we assume that the temperature is low enough so that the reactive flux focuses around a finite number of paths connecting the reactant and the product states these paths can be found in a systematic way by solving a hamilton jacobi equation for the so called maxflux functional we argue that the name maxflux is misleading it should be called the resistivity functional instead once the network of transition paths is found one can define an equivalent electric circuit and find the currents through each of its wires these currents give estimates of the reactive flux along the corresponding transition paths we test our approach on the problem of finding transition paths in the alanine dipeptide with two dihedral angles where the reactive current can be computed exactly the percentages of the reactive flux through each reactive channel given by our approach turn out to be in remarkable agreement with the exact ones we apply this approach to the problem of finding escape paths of a co molecule from a myoglobin protein we find a collection of exit locations and establish percentages of the reactive flux through each of them

high temperature limits for 1 1 dimensional directed polymer with heavy tailed disorder the directed polymer model at intermediate disorder regime was introduced by alberts khanin quastel ann probab 42 2014 1212 1256 it was proved that at inverse temperature n n with 1 4 1 4 the partition function centered appropriately converges in distribution and the limit is given in terms of the solution of the stochastic heat equation this result was obtained under the assumption that the disorder variables posses exponential moments but its universality was also conjectured under the assumption of six moments we show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments we also explain the behavior of the scaling exponent for the log partition function under different moment assumptions and values of

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

kpz line ensemble for each t ge 1 we construct an mathbb n indexed ensemble of random continuous curves with three properties 1 the lowest indexed curve is distributed as the time t hopf cole solution to the kardar parisi zhang kpz stochastic partial differential equation with narrow wedge initial data 2 the entire ensemble satisfies a resampling invariance which we call the mathbf h brownian gibbs property with mathbf h x e x 3 increments of the lowest indexed curve when centered by t 24 and scaled down vertically by t 1 3 and horizontally by t 2 3 remain uniformly absolutely continuous i e have tight radon nikodym derivatives with respect to brownian bridges as time t goes to infinity this construction uses as inputs the diffusion that o connell discovered ann probab 40 437 458 2012 in relation to the o connell yor semi discrete brownian polymer the convergence result of moreno flores et al in preparation of the lowest indexed curve of that diffusion to the solution of the kpz equation with narrow wedge initial data and the one point distribution formula proved by amir et al commun pure appl math 64 466 537 2011 for the solution of the kpz equation with narrow wedge initial data we provide four main applications of this construction 1 uniform as t goes to infinity brownian absolute continuity of the time t solution to the kpz equation with narrow wedge initial data even when scaled vertically by t 1 3 and horizontally by t 2 3 2 universality of the t 1 3 one point vertical fluctuation scale for the solution of the kpz equation with general initial data 3 concentration in the t 2 3 scale for the endpoint of the continuum directed random polymer 4 exponential upper and lower tail bounds for the solution at fixed time of the kpz equation with general initial data

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

point to line last passage percolation and the invariant measure of a system of reflecting brownian motions this paper proves an equality in law between the invariant measure of a reflected system of brownian motions and a vector of point to line last passage percolation times in a discrete random environment a consequence describes the distribution of the all time supremum of dyson brownian motion with drift a finite temperature version relates the point to line partition functions of two directed polymers with an inverse gamma and a brownian environment and generalises dufresne s identity our proof introduces an interacting system of brownian motions with an invariant measure given by a field of point to line log partition functions for the log gamma polymer

prestack migration by split step dsr the double square root dsr prestack migration equation though defined for depth variable velocity can be used to image media with strong velocity variations using a phase shift plus interpolation pspi or split step correction the split step method is based on applying a phase shift correction to the extrapolated wavefield a correction that attempts to compensate for the lateral velocity variations i show how to extend dsr prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the marmousi data set the split step dsr migration is very fast and offers excellent imaging results

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

nearest neighbor markov dynamics on macdonald processes abstract macdonald processes are certain probability measures on two dimensional arrays of interlacing particles introduced by borodin and corwin in 7 they are defined in terms of nonnegative specializations of the macdonald symmetric functions and depend on two parameters q t 0 1 our main result is a classification of continuous time nearest neighbor markov dynamics on the space of interlacing arrays that act nicely on macdonald processes the classification unites known examples of such dynamics and also yields many new ones when t 0 one dynamics leads to a new integrable interacting particle system on the one dimensional lattice which is a q deformation of the pushtasep long range tasep when q t the macdonald processes become the schur processes of okounkov and reshetikhin 41 in this degeneration we discover new robinson schensted type correspondences between words and pairs of young tableaux that govern some of our dynamics

a combined extended finite element and level set method for biofilm growth this paper presents a computational technique based on the extended finite element method xfem and the level set method for the growth of biofilms the discontinuous derivative enrichment of the standard finite element approximation eliminates the need for the finite element mesh to coincide with the biofilm fluid interface and also permits the introduction of the discontinuity in the normal derivative of the substrate concentration field at the biofilm fluid interface the xfem is coupled with a comprehensive level set update scheme with velocity extensions which makes updating the biofilm interface fast and accurate without need for remeshing the kinetics of biofilms are briefly given and the non linear strong and weak forms are presented the non linear system of equations is solved using a newton raphson scheme example problems including 1d and 2d biofilm growth are presented to illustrate the accuracy and utility of the method the 1d results we obtain are in excellent agreement with previous 1d results obtained using finite difference methods our 2d results that simulate finger formation and finger tip splitting in biofilms illustrate the robustness of the present computational technique copyright 2007 john wiley sons ltd

stationary correlations for the 1d kpz equation we study exact stationary properties of the one dimensional kardar parisi zhang kpz equation by using the replica approach the stationary state for the kpz equation is realized by setting the initial condition the two sided brownian motion bm with respect to the space variable developing techniques for dealing with this initial condition in the replica analysis we elucidate some exact nature of the height fluctuation for the kpz equation in particular we obtain an explicit representation of the probability distribution of the height in terms of the fredholm determinants furthermore from this expression we also get the exact expression of the space time two point correlation function

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

formulas for joint probabilities for the asymmetric simple exclusion process in earlier work the authors tracy c a and widom h integral formulas for the asymmetric simple exclusion process commun math phys 279 815 2008 obtained integral formulas for probabilities for a single particle in the asymmetric simple exclusion process here formulas are obtained for joint probabilities for several particles in the case of a single particle the derivation here is simpler than the one in the earlier work for one of its main results

integral formulas for the asymmetric simple exclusion process in this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process asep on the integer lattice mathbb z with nearest neighbor hopping rates p to the right and q 1 p to the left for the most part we consider an n particle system but for certain of these formulas we can take the n to infty limit first we obtain for the n particle system a formula for the probability of a configuration at time t given the initial configuration for this we use bethe ansatz ideas to solve the master equation extending a result of schutz for the case n 2 the main results of the paper derived from this are integral formulas for the probability for given initial configuration that the mth left most particle is at x at time t in one of these formulas we can take the n to infty limit and it gives the probability for an infinite system where the initial configuration is bounded on one side for the special case of the totally asymmetric simple exclusion process tasep our formulas reduce to the known ones

a level set approach for the solution of a state constrained optimal control problem state constrained optimal control problems for linear elliptic partial differential equations are considered the corresponding first order optimality conditions in primal dual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary interface between the active and inactive sets as optimization variable the new algorithm is based on the level set methodology the speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization encouraging numerical results attained by the new algorithm are reported on

computing minimal surfaces via level set curvature flow given a fixed closed curve or curves in r sp3 there is an associated problem of finding a surface with mean curvature zero which has that curve or curves as its boundary in this paper a new approach to numerically solving the minimal surface problem is introduced the surface is represented as a level set of a global function phi r sp3 to r the family of surfaces evolve in time according to mean curvature flow a solution is found when a steady state condition is achieved a new system of interpolatory boundary conditions are used to maintain the connection between the surface and the fixed boundary contour several computer surfaces are shown including examples of topological changes

robust and efficient upwind finite difference traveltime calculations in three dimensions first arrival traveltimes in complicated 3 d geologic media may be computed robustly and efficiently using an upwind finite difference solution of the 3 d eikonal equation an important application of this technique is computing traveltimes for imaging seismic data with 3 d prestack kirchhoff depth migration the method performs radial extrapolation of the three components of the slowness vector in spherical coordinates traveltimes are computed by numerically integrating the radial component of the slowness vector the original finite difference equations are recast into unitless forms that are more stable to numerical errors a stability condition adaptively determines the radial steps that are used to extrapolate computations are done in a rotated spherical coordinate system that places the small arc length regions of the spherical grid at the earth s surface z 0 plane this improves efficiency by placing large grid cells in the central regions of the grid where wavefields are complicated thereby

a kpz cocktail shaken not stirred the stochastic partial differential equation proposed nearly three decades ago by kardar parisi and zhang kpz continues to inspire intrigue and confound its many admirers here we i pay debts to heroic predecessors ii highlight additional experimentally relevant aspects of the recently solved 1 1 kpz problem iii use an expanding substrates formalism to gain access to the 3d radial kpz equation and lastly iv examining extremal paths on disordered hierarchical lattices set our gaze upon the fate of d infty kpz clearly there remains ample unexplored territory within the realm of kpz and for the hearty much work to be done especially in higher dimensions where numerical and renormalization group methods are providing a deeper understanding of this iconic equation

large deviation rate functions for the partition function in a log gamma distributed random potential we study right tail large deviations of the logarithm of the partition function for directed lattice paths in i i d random potentials the main purpose is the derivation of explicit formulas for the 1 1 dimensional exactly solvable case with log gamma distributed random weights along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

stochastic higher spin six vertex model and macdonald measures we prove an identity that relates the q laplace transform of the height function of a higher spin inhomogeneous stochastic six vertex model in a quadrant on one side and a multiplicative functional of a macdonald measure on the other the identity is used to prove the gue tracy widom asymptotics for two instances of the stochastic six vertex model via asymptotic analysis of the corresponding schur measures

information theoretic active polygons for unsupervised texture segmentation curve evolution models used in image segmentation and based on image region information usually utilize simple statistics such as means and variances hence can not account for higher order nature of the textural characteristics of image regions in addition the object delineation by active contour methods results in a contour representation which still requires a substantial amount of data to be stored for subsequent multimedia applications such as visual information retrieval from databases polygonal approximations of the extracted continuous curves are required to reduce the amount of data since polygons are powerful approximators of shapes for use in later recognition stages such as shape matching and coding the key contribution of this paper is the development of a new active contour model which nicely ties the desirable polygonal representation of an object directly to the image segmentation process this model can robustly capture texture boundaries by way of higher order statistics of the data and using an information theoretic measure and with its nature of the ordinary differential equations this new variational texture segmentation model is unsupervised since no prior knowledge on the textural properties of image regions is used another contribution in this sequel is a new polygon regularizer algorithm which uses electrostatics principles this is a global regularizer and is more consistent than a local polygon regularization in preserving local features such as corners

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions