contextual estimators of mixing probabilities for markov chain random fields abstract this paper discusses the estimation of proportions of classes from an image classification methods are compared with likelihood methods and the importance of contextual information is discussed the joint distribution of classes at neighbouring sites is modelled by a markov chain random field the class attributes are estimated from training sets and unclassified observations the effect of biased class means is reduced with a stochastic model of the bias contextual likelihood methods yield better results than non contextual methods

on modified factorizations for large scale linearly constrained optimization we consider the algebraic issues concerning the solution of general large scale linearly constrained nonlinear optimization problems particular attention is given to suitable methods for solving the linear systems that occur at each iteration of such methods the main issue addressed is how to ensure that a quadratic model of the objective function is positive definite in the null space of the constraints while neither adversely affecting the convergence of newton s method nor incurring a significant computational overhead numerical evidence to support the theoretical developments is provided

general β jacobi corners process and the gaussian free field we prove that the two dimensional gaussian free field describes the asymptotics of global fluctuations of a multilevel extension of the general jacobi random matrix ensembles our approach is based on the connection of the jacobi ensembles to a degeneration of the macdonald processes that parallels the degeneration of the macdonald polynomials to the heckman opdam hypergeometric functions of type a we also discuss the limit 2015 wiley periodicals inc

multilevel dyson brownian motions via jack polynomials we introduce multilevel versions of dyson brownian motions of arbitrary parameter beta 0 generalizing the interlacing reflected brownian motions of warren for beta 2 such processes unify beta corners processes and dyson brownian motions in a single object our approach is based on the approximation by certain multilevel discrete markov chains of independent interest which are defined by means of jack symmetric polynomials in particular this approach allows to show that the levels in a multilevel dyson brownian motion are intertwined at least for beta ge 1 and to give the corresponding link explicitly

parametric inference for imperfectly observed gibbsian fields this paper presents a maximum likelihood estimation method for imperfectly observed gibbsian fields on a finite lattice this method is an adaptation of the algorithm given in younes 28 presentation of the new algorithm is followed by a theorem about the limit of the second derivative of the likelihood when the lattice increases which is related to convergence of the method some practical remarks about the implementation of the procedure are eventually given

schur dynamics of the schur processes abstract we construct discrete time markov chains that preserve the class of schur processes on partitions and signatures one application is a simple exact sampling algorithm for q volume distributed skew plane partitions with an arbitrary back wall another application is a construction of markov chains on infinite gelfand tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite dimensional unitary group

shuffling algorithm for boxed plane partitions we introduce discrete time markov chains that preserve uniform measures on boxed plane partitions elementary markov steps change the size of the box from a b c to a 1 b 1 c or a 1 b 1 c algorithmic realization of each step involves o a b c operations one application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions trajectories of our markov chains can be viewed as random point configurations in the three dimensional lattice we compute the bulk limits of the correlation functions of the resulting random point process on suitable two dimensional sections the limiting correlation functions define a two dimensional determinantal point processes with certain gibbs properties

interlacing eigenvalues in time reversible markov chains for irreducible reversible finite state chains we observe that two sets of eigenvalues related to the transition rate matrix q 0 m and 1 m are interlaced so that 0 1 1 m m many quantities associated with l ta the distribution of the first passage time to a starting in steady state can be simply expressed in terms of these eigenvalues and the interlacing property can be exploited to obtain approximations there is a close connection between interlacing eigenvalues and completely monotone distributions as well as a representation for finite mixtures of exponential distributions which follow from this observation

interacting particle systems at the edge of multilevel dyson brownian motions abstract we study the joint asymptotic behavior of spacings between particles at the edge of multilevel dyson brownian motions when the number of levels tends to infinity despite the global interactions between particles in multilevel dyson brownian motions we observe a decoupling phenomenon in the limit the global interactions become negligible and only the local interactions remain the resulting limiting objects are interacting particle systems which can be described as brownian versions of certain totally asymmetric exclusion processes this is the first appearance of a particle system with local interactions in the context of general random matrix models

on implementing mehrotra s predictor corrector interior point method for linear programming mehrotra tech report 90 03 department of industrial engineering and management sciences northwestern university evanston il 1990 recently described a predictor corrector variant of the primal dual interior point algorithm for linear programming this paper describes a full implementation of this algorithm with extensions for solving problems with free variables and problems with bounds on primal variables computational results on the netlib test set are given to show that this new method almost always improves the performance of the primal dual algorithm and that the improvement increases dramatically as the size and complexity of the problem increases a numerical instability in using schur complements to remove dense columns is identified and a numerical remedy is given

improving monte carlo efficiency by increasing variance this paper compares the performances of two well known monte carlo procedures for estimating an unknown quantity as the size of the problem grows one method based on the standard monte carlo approach generates k i i d data points the other derives its data from a single k step sample path generated by a markov chain the paper gives necessary and sufficient conditions for the markov chain approach to perform more efficiently than the standard monte carlo approach does moreover it identifies circumstances under which this better efficiency grows with increasing problem size this improved efficiency comes from reduced sample generating and function evaluating costs in the markov chain approach that more than compensate for the increased variance of the estimator that the markov chain sampling approach induces when compared to the standard monte carlo approach several examples illustrate how the benefits arise one also demonstrates a case in which the standard monte carlo approach becomes increasingly preferred as the problem size grows

commutation relations and markov chains it is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain markov chains examples studied include random walk on irreducible representations a local random walk on partitions whose stationary distribution is the ewens distribution and some birth death chains

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

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

an implementation of linear and nonlinear multicommodity network flows abstract this work presents a new code for solving the multicommodity network flow problem with a linear or nonlinear objective function considering additional linear side constraints that link arcs of the same or different commodities for the multicommodity network flow problem through primal partitioning the code implements a specialization of murtagh and saunders strategy of dividing the set of variables into basic nonbasic and superbasic several tests are reported using random problems obtained from different network generators and real problems arising from the fields of long and short term hydrothermal scheduling of electricity generation and traffic assignment with sizes of up to 150000 variables and 45 000 constraints the performance of the code developed is compared to that of alternative methodologies for solving the same problems a general purpose linear and nonlinear constrained optimization code a specialised linear multicommodity network flow code and a primal dual interior point code

the augmented system variant of ipms in two stage stochastic linear programming computation abstract the application of interior point methods ipm to solve the deterministic equivalent of two stage stochastic linear programming problems is a known and natural idea experiments have proved that among the interior point methods the augmented system approach gives the best performance on these problems however most of their implementations encounter numerical difficulties in certain cases which can result in loss of efficiency we present a new approach for the decomposition of the augmented system which automatically exploits the special behavior of the problems we show that the suggested approach can be implemented in a fast and numerically robust way by solving a number of large scale two stage stochastic linear programming problems the comparison of our solver with fo1aug which is considered as a state of the art augmented system implementation of interior point methods is also given

future independent times and markov chains a random timet is a future independent time for a markov chain x n 0 ift is independent of x t n n 0 and if x t n n 0 is a markov chain with initial distribution and the same transition probabilities as x n 0 this concept is used with the conditional stationary measure to give a new and short proof of the basic limit theorem of markov chains improving somewhat the result in the null recurrent case

markov processes on the path space of the gelfand tsetlin graph and on its boundary we construct a four parameter family of markov processes on infinite gelfand tsetlin schemes that preserve the class of central gibbs measures any process in the family induces a feller markov process on the infinite dimensional boundary of the gelfand tsetlin graph or equivalently the space of extreme characters of the infinite dimensional unitary group u the process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on u posed in olshanski 2003 44 as was shown in borodin and olshanski 2005 11 this measure can also be described as a determinantal point process with a correlation kernel expressed through the gauss hypergeometric function

a new class of preconditioners for large scale linear systems from interior point methods for linear programming a new class of preconditioners for the iterative solution of the linear systems arising from interior point methods is proposed for many of these methods the linear systems are symmetric and indefinite the system can be reduced to a system of normal equations which is positive definite we show that all preconditioners for the normal equations system have an equivalent for the augmented system while the opposite is not true the new class of preconditioners works better near a solution of the linear programming problem when the matrices are highly ill conditioned the preconditioned system can be reduced to a positive definite one the techniques developed for a competitive implementation are rather sophisticated since the subset of columns is not known a priori the new preconditioner applied to the conjugate gradient method compares favorably with the cholesky factorization approach on large scale problems whose cholesky factorization contains too many nonzero entries

the mathematics of mixing things up how long should a markov chain monte carlo algorithm be run using examples from statistical physics ehrenfest urn ising model hard discs as well as card shuffling this tutorial paper gives an overview of a body of mathematical results that can give useful answers to practitioners viz seven shuffles suffice for practical purposes it points to new techniques path coupling geometric inequalities and harris recurrence the discovery of phase transitions in mixing times the cutoff phenomenon is emphasized

a 2 1 dimensional growth process with explicit stationary measures we introduce a class of 2 1 2 1 dimensional stochastic growth processes that can be seen as irreversible random dynamics of discrete interfaces irreversible means that the interface has an average nonzero drift interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice the model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting hammersley processes when the dynamical asymmetry parameter p q p q equals zero the infinite volume gibbs measures with given slope are stationary and reversible when p qp q are not reversible any more but remarkably they are still stationary in such stationary states we find that the average height function at any given point xx grows linearly with time tt with a nonzero speed eqx t e hx t hx 0 v teqx t e hx t hx 0 v t while the typical fluctuations of qx t qx t are smaller than any power of tt as t t in the totally asymmetric case of p 0 q 1p 0 q 1 and on the hexagonal lattice the dynamics coincides with the anisotropic kpz growth model introduced by a borodin and p l ferrari in j stat mech theory exp 2009 2009 p02009 comm math phys 325 603 684 for a suitably chosen integrable initial condition that is very far from the stationary state they were able to determine the hydrodynamic limit and a clt for interface fluctuations on scale logt log t exploiting the fact that in that case certain space time height correlations can be computed exactly in the same setting they proved that asymptotically for t t the local statistics of height fluctuations tends to that of a gibbs state which led to the prediction that gibbs states should be stationary

convergence of a class of inexact interior point algorithms for linear programs we present a convergence analysis for a class of inexact infeasible interior point methods for solving linear programs the main feature of inexact methods is that the linear systems defining the search direction at each interior point iteration need not be solved to high accuracy more precisely we allow that these linear systems are only solved to a moderate accuracy in the residual but no assumptions are made on the accuracy of the search direction in the search space in particular our analysis does not require that feasibility is maintained even if the initial iterate happened to be a feasible solution of the linear program we also present a few numerical examples to illustrate the effect of using inexact search directions on the number of interior point iterations

three dimensional gaussian fluctuations of non commutative random surfaces along time like paths abstract we construct a continuous time non commutative markov operator on u gl n with corresponding morphisms u gl n end l 2 u n at any fixed times 0 t 1 t 2 this is an analog of a continuous time non commutative markov operator on the group von neumann algebra v n u n constructed in 16 and is a variant of discrete time non commutative random walks on u gl n 2 9 it is also shown that when restricting to the gelfand tsetlin subalgebra of u gl n the non commutative markov operator matches a 2 1 dimensional random surface model introduced in 7 as an application it is then proved that the moments converge to an explicit gaussian field along time like paths combining with 7 which showed convergence to the gaussian free field along space like paths this computes the entire three dimensional gaussian field in particular it matches a gaussian field from eigenvalues of random matrices 5

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

convergence properties of an augmented lagrangian algorithm for optimization with a combination of general equality and linear constraints we consider the global and local convergence properties of a class of augmented lagrangian methods for solving nonlinear programming problems in these methods linear and more general constraints are handled in different ways the general constraints are combined with the objective function in an augmented lagrangian the iteration consists of solving a sequence of subproblems in each subproblem the augmented lagrangian is approximately minimized in the region defined by the linear constraints a subproblem is terminated as soon as a stopping condition is satisfied the stopping rules that we consider here encompass practical tests used in several existing packages for linearly constrained optimization our algorithm also allows different penalty parameters to be associated with disjoint subsets of the general constraints in this paper we analyze the convergence of the sequence of iterates generated by such an algorithm and prove global and fast linear convergence as well as show that potentially troublesome penalty parameters remain bounded away from zero

a simulation based estimator for hidden markov random fields an estimator for estimating the parameters of a markov random field x from inaccurate observations is introduced the author considers first a markov gibbs random field x x sub i j on a lattice l i j i 1 2 n j 1 2 m the marginal distributions of x sub i j x sub i u j v u v 1 0 1 are first estimated from an image then random fields x are simulated with the probability of x sub i u j v b nearly equal to the estimate of p x sub i j x sub i u b a simulation method similar to the gibbs sampler is used the parameters of the markov random field model are estimated from the x s with the pseudolikelihood method

bayes smoothing algorithms for segmentation of binary images modeled by markov random fields a new image segmentation algorithm is presented based on recursive bayes smoothing of images modeled by markov random fields and corrupted by independent additive noise the bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel given the total noisy image in a recursive way the a posteriori distribution together with a criterion of optimality then determine a bayes estimate of the scene the algorithm presented is an extension of a 1 d bayes smoothing algorithm to 2 d and it gives the optimum bayes estimate for the scene value at each pixel computational concerns in 2 d however necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm in particular the scene noiseless image is modeled as a markov mesh random field a special class of markov random fields and the bayes smoothing algorithm is applied on overlapping strips horizontal vertical of the image consisting of several rows columns it is assumed that the signal scene values vector sequence along the strip is a vector markov chain since signal correlation in one of the dimensions is not fully used along the edges of the strip estimates are generated only along the middle sections of the strips the overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image the bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel discrete or continuous random variables

stochastic linear programs with restricted recourse abstract stochastic programs with recourse provide an effective modeling paradigm for sequential decision problems with uncertain or noisy data when uncertainty can be modeled by a discrete set of scenarios in two stage problems the decision variables are partitioned into two groups a set of structural first stage decisions and a set of second stage recourse decisions the structural decisions are scenario invariant but the recourse decisions are scenario dependent and can vary substantially across scenarios in several applications it is important to restrict the variability of recourse decisions across scenarios or to investigate the tradeoffs between the stability of recourse decisions and expected cost of a solution we present formulations of stochastic programs with restricted recourse that trade off recourse stability with expected cost the models generate a sequence of solutions to which recourse robustness is progressively enforced via parameterized satisficing constraints we investigate the behavior of the models on several test cases and examine the performance of solution procedures based on the primal dual interior point method

an iterative gibbsian technique for reconstruction of m ary images abstract the reconstruction of m ary images corrupted by independent noise is treated the original image is modeled by a markov random field mrf whose parameters are unknown likewise the probabilistic structure of the noise is unknown this paper presents an iterative procedure which performs the parameter estimation and image reconstruction tasks at the same time the procedure that we call gibbsian em algorithm is a generalization to the mrf context of a general algorithm known as the em algorithm used to approximate maximum likelihood estimates for incomplete data problems a number of experiments are presented in the case of gaussian noise and binary noise showing that the gibbsian em algorithm is useful and effective for image reconstruction and segmentation

error analysis of crop acreage estimation using satellite data the problem of crop acreage estimation using satellite data is discussed the bias and variance of a crop estimate in an area segment obtained from the classification of its multispectral sensor data are derived as functions of the means variances and covariances of error rates linear discriminant analysis and class proportion estimation for the two class case have been extended to include a third class of measurement units where these units are mixed on the ground special attention is given to the investigation of mislabeling in training samples and its effect on crop proportion estimation it is shown that the bias and variance of the estimate of a crop acreage proportion increase as the disparity in mislabeling rates of the two classes increases the effect of mixed pixels on the bias and variance of the proportion estimate is also investigated

on the integrability of zero range chipping models with factorized steady states the conditions of the integrability of general zero range chipping models with factorized steady states which were proposed in evans et al 2004 j phys a math gen 37 l275 are examined we find a three parametric family of hopping probabilities for the models solvable by the bethe ansatz which includes most of known integrable stochastic particle models as limiting cases the solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations which is proved as a byproduct we use the bethe ansatz to solve an eigenproblem for the transition matrix of the markov process on its basis we conjecture an integral formula for the green function of the evolution operator for the model on an infinite lattice and derive the bethe equations for the spectrum of the model on a ring

the factorability of symmetric matrices and some implications for statistical linear models abstract general conditions where a symmetric matrix is factorable by cholesky decomposition are described while numerical stability is a remaining issue whenever the cholesky decomposition is used to factor indefinite matrices the existence of such factors is demonstrated for matrix structures that are commonly found in statistics kalman filtering for example is rediscovered in the cholesky decomposition of an indefinite matrix moreover the cholesky decomposition uniquely defines the likelihood function in linear statistical models and this includes situations when the variance matrix is singular or when the cholesky decomposition does not run to completion alternative methods of likelihood evaluation which may involve for example the bunch parlett factorization are available only when the cholesky decomposition exists suggestions are made for computing an adaptive precision cholesky decomposition when numerical stability is an issue

unsupervised context estimation in a mesh of pattern classes for image recognition abstract this paper defines a wide class of models for the statistical dependency of class labels context in a neighborhood within an image a computationally efficient closed form estimator for these context defining parameters is developed the estimator is unbiased convergent in the mean square sense and can be recursively implemented parameters of several practical context models within the class of models are estimated as functions of the estimates of the parameters in the general model the appropriate model can also be decided with the help of the context parameter estimates applications of the estimator in existing context classifiers are mentioned a limitation of the estimator is also pointed out iterative approaches for decision making in markov models are outlined to highlight the use of our estimator to ensure the convergence of iterative decisions to a local optimum

random forests and networks analysis wilson proceedings of the twenty eight annual acm symposium on the theory of computing pp 296 303 1996 in the 1990s described a simple and efficient algorithm based on loop erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph this algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking in recent works avena and gaudilliere in a proof of the transfer current theorem in absence of reversibility in stat probab lett 142 17 22 2018 avena and gaudilliere in j theor probab 2017 https doi org 10 1007 s10959 017 0771 3 avena et al in approximate and exact solutions of intertwining equations though random spanning forests 2017 arxiv 1702 05992v1 avena et al in intertwining wavelets or multiresolution analysis on graphs through random forests 2017 arxiv 1707 04616 to appear in acha 2018 we focused on applications of spanning rooted forests on finite graphs the resulting main conclusions are reviewed in this paper by collecting related theorems algorithms heuristics and numerical experiments a first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications 1 a random walk based notion of well distributed points in a graph 2 a framework to describe metastable like dynamics in finite settings by means of markov intertwining dualities 3 coarse graining schemes for networks and associated processes 4 wavelets like pyramidal algorithms for graph signals

siegmund duality with applications to the neutral moran model conditioned on never being absorbed we first consider the classical neutral moran model with two alleles whose fate is either to become extinct or to reach fixation we study an ergodic version of the moran model obtained by conditioning it to never hit the boundaries making use of a doob transform we call it the recurrent moran model we show that the siegmund dual of the recurrent moran process exists and is a substochastic birth and death chain conditioning this process to exit in its natural absorbing state we construct a process with a unique absorbing state which is intertwined to the original recurrent moran process the time needed for the intertwined process to first hit its absorbing state is related to the time needed to reach stationarity for the recurrent moran process using spectral information on the intertwined chain we extract limiting information on this first hitting time that shows that there is no abrupt relaxation to equilibrium for the recurrent moran chain this makes use of the relation between duality and intertwining and strong stationary times other related transition times of the recurrent moran chain are also briefly investigated namely the first return time to the ground state and the expected time needed to move from one end to the other end of the state space

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

symmetric indefinite systems for interior point methods we present a unified framework for solving linear and convex quadratic programs via interior point methods at each iteration this method solves an indefinite system whose matrix is left begin array 20 c d 2 a t a 0 end array right instead of reducing to obtain the usualad 2 a t system this methodology affords two advantages 1 it avoids the fill created by explicitly forming the productad 2 a t whena has dense columns and 2 it can easily be used to solve nonseparable quadratic programs since it requires only thatd be symmetric we also present a procedure for converting nonseparable quadratic programs to separable ones which yields computational savings when the matrix of quadratic coefficients is dense

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

maximum likelihood discriminant analysis on the plane using a markovian model of spatial context abstract a method is presented for utilising spatial information in performing discriminant analysis on multivariate data at each point on a regular lattice as for example with landsat the spatial information is modelled as a particularly simple markov random field the discriminant analysis is based on the maximum likelihood method the results are compared with alternatives by monte carlo simulation and seem to be encouraging

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

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

the one dimensional kpz equation and its universality class our understanding of the one dimensional kpz equation alias noisy burgers equation has advanced substantially over the past 5 years we provide a non technical review where we limit ourselves to the stochastic pde and lattice type models approximating it

computational study of a family of mixed integer quadratic programming problems we present computational experience with a branch and cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables such problems arise in financial applications the algorithm solves the largest real life problems in a few minutes of run time

on eigenfunctions of markov processes on trees we begin by studying the eigenvectors associated to irreducible finite birth and death processes showing that the i nontrivial eigenvector i admits a succession of i decreasing or increasing stages each of them crossing zero imbedding naturally the finite state space into a continuous segment one can unequivocally define the zeros of i which are interlaced with those of i 1 these kind of results are deduced from a general investigation of minimax multi sets dirichlet eigenproblems which leads to a direct construction of the eigenvectors associated to birth and death processes this approach can be generically extended to eigenvectors of markov processes living on trees this enables to reinterpret the eigenvalues and the eigenvectors in terms of the previous dirichlet eigenproblems and a more general conjecture is presented about related higher order cheeger inequalities finally we carefully study the geometric structure of the eigenspace associated to the spectral gap on trees

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 qmr based interior point algorithm for solving linear programs a new approach for the implementation of interior point methods for solving linear programs is proposed its main feature is the iterative solution of the symmetric but highly indefinite 2 2 block systems of linear equations that arise within the interior point algorithm these linear systems are solved by a symmetric variant of the quasi minimal residual qmr algorithm which is an iterative solver for general linear systems the symmetric qmr algorithm can be combined with indefinite preconditioners which is crucial for the efficient solution of highly indefinite linear systems yet it still fully exploits the symmetry of the linear systems to be solved to support the use of the symmetric qmr iteration a novel stable reduction of the original unsymmetric 3 3 block systems to symmetric 2 2 block systems is introduced and a measure for a low relative accuracy for the solution of these linear systems within the interior point algorithm is proposed some indefinite preconditioners are discussed finally we report results of a few preliminary numerical experiments to illustrate the features of the new approach

a variation on karmarkar s algorithm for solving linear programming problems the algorithm described here is a variation on karmarkar s algorithm for linear programming it has several advantages over karmarkar s original algorithm in the first place it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function the minimum value of the objective function does not have to be known in advance secondly in the absence of degeneracy the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero this makes it possible to identify an optimal basis before the algorithm converges

solving symmetric indefinite systems in an interior point method for linear programming we describe an implementation of a primal dual path following method for linear programming that solves symmetric indefinite augmented systems directly by bunch parlett factorization rather than reducing these systems to the positive definite normal equations that are solved by cholesky factorization in many existing implementations the augmented system approach is seen to avoid difficulties of numerical instability and inefficiency associated with free variables and with dense columns in the normal equations approach solving the indefinite systems does incur an extra overhead whose median is about 40 in our tests but the augmented system approach proves to be faster for a minority of cases in which the normal equations have relatively dense cholesky factors a detailed analysis shows that the augmented system factorization is reliable over a fairly large range of the parameter settings that control the tradeoff between sparsity and numerical stability

stationary states of crystal growth in three dimensions a new markov process describing crystal growth in three dimensions is introduced states of the process are configurations of the crystal surface which has a terrace edge kink structure the states are continuous along edges but discrete across edges in accordance with the very different rates for the two types of captures of particles stationary distributions describing steady crystal growth are found in general to our knowledge these are the first examples of stationary distributions for layered crystal growth in three dimensions the steady growth rate and other quantities are obtained explicitly for two interacting edges for many interacting edges growth behavior is determined a in various asymptotic regimes including thermodynamic limits b via simulations and c using series cluster expansions in the slope of the surface the first three coefficients being computed the theoretical growth rates show a marked dependence on surface dimensions this may contribute to the size dependence and dispersion in the observed growth rate of small crystals

bias associated with the discriminant analysis approach to the estimation of mixing proportions abstract the estimation of the mixing proportion 1 in which the first of two multivariate normal groups occur is considered on the basis of a sample drawn from a mixture of them under the model studied there are also data of known origin available from each of the groups for the formation of a discriminant rule r the bias of the estimator of 1 based on the proportion of the mixture data assigned to the first group by r is investigated also the case where the data of known origin are taken to be correlated is considered

computational experience with a primal dual interior point method for linear programming abstract a new comprehensive implementation of a primal dual algorithm for linear programming is described it allows for easy handling of simple bounds on the primal variables and incorporates free variables which have not previously been included in a primal dual implementation we discuss in detail a variety of computational issues concerning the primal dual implementation and barrier methods for linear programming in general we show that in a certain way lustig s method for obtaining feasibility is equivalent to newton s method this demonstrates that the method is in some sense the natural way to reduce infeasibility the role of the barrier parameter in computational practice is studied in detail numerical results are given for the entire expanded netlib test set for the basic algorithm and its variants as well as version 5 3 of minos

proof of aldous spectral gap conjecture aldous spectral gap conjecture asserts that on any graph the random walk process and the random transposition or interchange process have the same spectral gap we prove the conjecture using a recursive strategy the approach is a natural extension of the method already used to prove the validity of the conjecture on trees the novelty is an idea based on electric network reduction which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates the proof of the latter inequality uses suitable coset decompositions of the associated matrices with rows and columns indexed by permutations

mixing time for random walk on supercritical dynamical percolation we consider dynamical percolation on the d dimensional discrete torus znd of side length n where each edge refreshes its status at rate n 1 2 to be open with probability p we study random walk on the torus where the walker moves at rate 1 2d along each open edge in earlier work of two of the authors with a stauffer it was shown that in the subcritical case p pc zd the mixing time is n2 1 here the implied constants depend only on d and p we prove a quenched and hence annealed version of this conjecture up to a poly logarithmic factor under the assumption p 1 2 when p 0 we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times the latter implies sharp up to poly logarithmic factors upper bounds on exit times of large balls throughout the supercritical regime our proofs are based on percolation results e g the grimmett marstrand theorem and an analysis of the volume biased evolving set process the key point is that typically the evolving set has a substantial intersection with the giant percolation cluster at many times this allows us to use precise isoperimetric properties of the cluster due to g pete to infer rapid growth of the evolving set which in turn yields the upper bound on the mixing time

total variation cutoff in birth and death chains the cutoff phenomenon describes a case where a markov chain exhibits a sharp transition in its convergence to stationarity diaconis proc natl acad sci usa 93 4 1659 1664 1996 surveyed this phenomenon and asked how one could recognize its occurrence in families of finite ergodic markov chains peres american institute of mathematics aim research workshop palo alto http www aimath org wwn mixingtimes 2004 noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing time and spectral gap tends to infinity and conjectured that in many settings this condition should also be sufficient diaconis and saloff coste ann appl probab 16 4 2098 2122 2006 verified this conjecture for continuous time birth and death chains started at an endpoint with convergence measured in separation it is natural to ask whether the conjecture holds for these chains in the more widely used total variation distance in this work we confirm the above conjecture for all continuous time or lazy discrete time birth and death chains with convergence measured via total variation distance namely if the product of the mixing time and spectral gap tends to infinity the chains exhibit cutoff at the maximal hitting time of the stationary distribution median with a window of at most the geometric mean between the relaxation time and mixing time in addition we show that for any lazy or continuous time birth and death chain with stationary distribution the separation 1 p t x y y is maximized when x y are the endpoints together with the above results this implies that total variation cutoff is equivalent to separation cutoff in any family of such chains