extracting macroscopic stochastic dynamics model problems the purpose of this work is to shed light on an algorithm designed to extract effective macroscopic models from detailed microscopic simulations the particular algorithm we study is a recently developed transfer operator approach due to schutte et al 20 the investigations involve the formulation and subsequent numerical study of a class of model problems the model problems are ordinary differential equations constructed to have the property that when projected onto a low dimensional subspace the dynamics is approximately that of a stochastic differential equation exhibiting a finite state space markov chain structure the numerical studies show that the transfer operator approach can accurately extract finite state markov chain behavior embedded within high dimensional ordinary differential equations in so doing the studies lend considerable weight to existing applications of the algorithm to the complex systems arising in applications such as molecular dynamics the algorithm is predicated on the assumption of markovian input data further numerical studies probe the role of memory effects although preliminary these studies of memory indicate interesting avenues for further development of the transfer operator methodology

a direct approach to conformational dynamics based on hybrid monte carlo recently a novel concept for the computation ofessentialfeatures of the dynamics of hamiltonian systems such as molecular dynamics has been proposed the realization of this concept had been based on subdivision techniques applied to the frobenius perron operator for the dynamical system the present paper suggests an alternative but related concept that merges the conceptual advantages of the dynamical systems approach with the appropriate statistical physics framework this approach allows us to define the phrase conformation in terms of the dynamical behavior of the molecular system and to characterize the dynamical stability of conformations in a first step the frequency of conformational changes is characterized in statistical terms leading to the definition of some markov operatortthat describes the corresponding transition probabilities within the canonical ensemble in a second step a discretization oftvia specific hybrid monte carlo techniques is shown to lead to astochasticmatrixp with these theoretical preparaions an identification algorithm for conformations to be presented in a later paper is applicable it is demonstrated that the discretization oftcan be restricted to few essential degrees of freedom so that the combinatorial explosion of discretization boxes is prevented and biomolecular systems can be attacked numerical results for the n pentane molecule and the triribonucleotide adenylyl 3 5 cytidylyl 3 5 cytidin are given and interpreted

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

equivariant quantum schubert polynomials abstract we establish an equivariant quantum giambelli formula for partial flag varieties the answer is given in terms of a specialization of universal double schubert polynomials along the way we give new proofs of the presentation of the equivariant quantum cohomology ring as well as graham positivity of the structure constants in equivariant quantum schubert calculus

challenges in climate science and contemporary applied mathematics this article discusses the challenges in climate science from the emerging viewpoint of stochastic statistical properties of turbulent dynamical systems the mathematical topics discussed here include empirical information theory fluctuation dissipation theorems reduced order stochastic modeling and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems the applied mathematics topics include the emerging development of multiscale algorithms for filtering data assimilation and superparametrization for climate science and other problems in science and engineering as well as suitable unambiguous mathematical test problems for their behavior interesting contemporary research directions and specific open problems are mentioned throughout the article the perspective here should also be useful for applications to other complex dynamical systems arising in neural science material science and environmental mechanical engineering 2011 wiley periodicals inc

switching regression models and fuzzy clustering a family of objective functions called fuzzy c regression models which can be used too fit switching regression models to certain types of mixed data is presented minimization of particular objective functions in the family yields simultaneous estimates for the parameters of c regression models together with a fuzzy c partitioning of the data a general optimization approach for the family of objective functions is given and corresponding theoretical convergence results are discussed the approach is illustrated by two numerical examples that show how it can be used to fit mixed data to coupled linear and nonlinear models

stanley symmetric functions and quiver varieties abstract we prove that the coefficients obtained when stanley symmetric functions are expanded in the basis of schur functions are special cases of the coefficients in the formula for quiver varieties given by the author and w fulton 1999 invent math 135 665 687 and we discuss the relations of this with a conjectured littlewood richardson rule for these quiver coefficients in addition we generalize a factorization formula for schubert polynomials of a product of two permutations

tikhonov regularization and the l curve for large discrete ill posed problems discretization of linear inverse problems generally gives rise to very ill conditioned linear systems of algebraic equations typically the linear systems obtained have to be regularized to make the computation of a meaningful approximate solution possible tikhonov regularization is one of the most popular regularization methods a regularization parameter specifies the amount of regularization and in general an appropriate value of this parameter is not known a priori we review available iterative methods and present new ones for the determination of a suitable value of the regularization parameter by the l curve criterion and the solution of regularized systems of algebraic equations

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

chern class formulas for quiver varieties our goal in this paper is to prove a formula for the general degeneracy locus xr associated to an oriented quiver of type an if we are given a sequence of vector bundles and vector bundle maps e0 1 e1 2 e2 y y eny1 n en on an algebraic variety x and a collection r rij 0 i j n of non negative integers there is a degeneracy locus xr xr e xr e de ned by xr fx 2 x j rank ei x ej x rij 8 i jg 1 1

divisors of secant planes to curves abstract inside the symmetric product of a very general curve we consider the codimension one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension we compute the classes of such divisors and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve by letting the moduli of the curve vary we study more generally the classes of the related divisors inside the moduli space of stable pointed curves

satisfiability in many valued sentential logic is np complete abstract cook s np completeness theorem is extended to all many valued sentential logics including the infinite valued calculus of ukasiewicz

principal dynamical components a procedure is proposed for a dimension reduction in time series similarly to principal components the procedure seeks a low dimensional manifold that minimizes information loss unlike principal components however the procedure involves dynamical considerations through the proposal of a predictive dynamical model in the reduced manifold hence the minimization of the uncertainty is not only over the choice of a reduced manifold as in principal components but also over the parameters of the dynamical model as in autoregressive analysis and principal interaction patterns further generalizations are provided to nonautonomous and non markovian scenarios which are then applied to historical sea surface temperature data 2012 wiley periodicals inc

f regularity of large schubert varieties let g denote a connected reductive algebraic group over an algebraically closed field k and let x denote a projective g g equivariant embedding of g the large schubert varieties in x are the closures of the double cosets bgb where b denotes a borel subgroup of g and g g we prove that these varieties are globally f regular in positive characteristic resp of globally f regular type in characteristic 0 as a consequence the large schubert varieties are normal and cohen macaulay 1 introduction the class of globally f regular varieties was introduced by smith in 21 these are projective algebraic varieties in positive character istics such that all the ideals in their homogeneous coordinate rings are tightly closed the globally f regular varieties and their analogues in characteristic 0 the varieties of globally f regular type have remarkable properties e g they are normal and cohen macaulay and the higher cohomology groups of all nef invertible sheaves are trivial examples of globally f regular varieties include the projective toric varieties prop 6 4 in 21 in this note we obtain the global f regularity of a wider class of varieties with algebraic group action the g g equivariant projective embeddings of any connected reductive group g and the closures in any such embedding of the double cosets bgb where b denotes a borel subgroup of g and g g is arbitrary by the bruhat decomposition these large schubert varieties are parametrized by the weyl group of g examples are the closures of parabolic subgroups we also show that the large schubert varieties are of globally f regular type in characteristic 0 for this we exploit the close relation between global f regularity and frobe nius splitting established in 21 and the frobenius splitting properties of the large schubert varieties proved in 5 and 19 another key ingredient is the global f regularity of the flag varieties and their schubert varieties 12 note that unlike for schubert varieties no desingularization of large schubert varieties is known in general this makes our arguments somewhat indirect as a consequence of our result the large schubert varieties in any equivariant embedding of g are normal and cohen macaulay this was first proved in the case of the canonical compactification of a semisimple adjoint group by frobenius

presenting the cohomology of a schubert variety we extend the short presentation due to borel 53 of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its schubert varieties our result is stated in a root system uniform manner by introducing the essential set of a coxeter group element generalizing and giving a new characterization of fulton 92 s definition for permutations further refinements are obtained in type a

schubert polynomials and classes of hessenberg varieties abstract regular semisimple hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory numerical analysis representation theory algebraic geometry and combinatorics we give a giambelli formula expressing the classes of regular semisimple hessenberg varieties in terms of chern classes in fact we show that the cohomology class of each regular semisimple hessenberg variety is the specialization of a certain double schubert polynomial giving a natural geometric interpretation to such specializations we also decompose such classes in terms of the schubert basis for the cohomology ring of the flag variety the coefficients obtained are nonnegative and we give closed combinatorial formulas for the coefficients in many cases we introduce a closely related family of schemes called regular nilpotent hessenberg schemes and use our results to determine when such schemes are reduced

local model uncertainty and incomplete data bias with discussion problems of the analysis of data with incomplete observations are all too familiar in statistics they are doubly difficult if we are also uncertain about the choice of model we propose a general formulation for the discussion of such problems and develop approximations to the resulting bias of maximum likelihood estimates on the assumption that model departures are small loss of efficiency in parameter estimation due to incompleteness in the data has a dual interpretation the increase in variance when an assumed model is correct the bias in estimation when the model is incorrect examples include non ignorable missing data hidden confounders in observational studies and publication bias in meta analysis doubling variances before calculating confidence intervals or test statistics is suggested as a crude way of addressing the possibility of undetectably small departures from the model the problem of assessing the risk of lung cancer from passive smoking is used as a motivating example

subword complexes in coxeter groups abstract let be a coxeter system an ordered list of elements in and an element in determine a subword complex as introduced in knutson and miller ann of math 2 2003 to appear subword complexes are demonstrated here to be homeomorphic to balls or spheres and their hilbert series are shown to reflect combinatorial properties of reduced expressions in coxeter groups two formulae for double grothendieck polynomials one of which appeared in fomin and kirillov proceedings of the sixth conference in formal power series and algebraic combinatorics dimacs 1994 pp 183 190 are recovered in the context of simplicial topology for subword complexes some open questions related to subword complexes are presented

simple bratteli diagrams with a godel incomplete c equivalence problem an abstract simplicial complex is a finite family of subsets of a finite set closed under subsets every abstract simplicial complex c naturally determines a bratteli diagram and a stable af algebra a c consider the following problem input a pair of abstract simplicial complexes c and c question is a c isomorphic to a c we show that this problem is godel incomplete i e it is recursively enumerable but not decidable this result is in sharp contrast with the recent decidability result by bratteli jorgensen kim and roush for the isomorphism problem of stable af algebras arising from the iteration of the same positive integer matrix for the proof we use a combinatorial variant of the de concini procesi theorem for toric varieties together with the baker beynon duality theory for lattice ordered abelian groups markov s undecidability result and elliott s classification theory for af algebras

word problems in elliott monoids abstract algorithmic issues concerning elliott local semigroups are seldom considered in the literature although these combinatorial structures completely classify af algebras in general the addition operation of an elliott local semigroup is partial but for every af algebra b whose murray von neumann order of projections is a lattice this operation is uniquely extendible to the addition of an involutive monoid e b let m 1 be the farey af algebra introduced by the present author in 1988 and rediscovered by f boca in 2008 the freeness properties of the involutive monoid e m 1 yield a natural word problem for every af algebra b with singly generated e b because b is automatically a quotient of m 1 given two formulas and in the language of involutive monoids the problem asks to decide whether and code the same equivalence of projections of b this mimics the classical definition of the word problem of a group presented by generators and relations we show that the word problem of m 1 is solvable in polynomial time and so is the word problem of the behncke leptin algebras a n k and of the effros shen algebras f for 0 1 q a real algebraic number or 1 e we construct a quotient of m 1 having a godel incomplete word problem and show that no primitive quotient of m 1 is godel incomplete

farey stellar subdivisions ultrasimplicial groups and k0 of af c algebras abstract the theory will i believe admit of being extended to solid reticulations formed by the intersections of three systems of equidistant parallel planes determinants of the third order between the three coordinates of successive points replacing the pq p q of the plane theory

double schubert polynomials for the classical groups abstract for each infinite series of the classical lie groups of type b c or d we construct a family of polynomials parametrized by the elements of the corresponding weyl group of infinite rank these polynomials represent the schubert classes in the equivariant cohomology of the appropriate flag variety they satisfy a stability property and are a natural extension of the single schubert polynomials of billey and haiman which represent non equivariant schubert classes they are also positive in a certain sense and when indexed by maximal grassmannian elements or by the longest element in a finite weyl group these polynomials can be expressed in terms of the factorial analogues of schur s q or p functions defined earlier by ivanov

identification of almost invariant aggregates in reversible nearly uncoupled markov chains the topic of the present paper has been motivated by a recent computational approach to identify metastable chemical conformations and patterns of conformational changes within molecular systems after proper discretization such conformations show up as almost invariant aggregates in reversible nearly uncoupled markov chains numcs most of the former work on this subject treated the direct problem given the aggregates analyze the loose coupling in connection with the computation of the stationary distribution aggregation disaggregation techniques in contrast to that the present paper focuses on the inverse problem given the system as a whole identify the almost invariant aggregates together with the small probabilities of transitions between them a robust algorithm is worked out on the basis of some detailed perturbation analysis and illustrated at a simple molecular system

response and sensitivity using markov chains dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study how this affects their statistical properties a prominent real life example of this class of problems is the investigation of climate response to perturbations in this respect it is crucial to determine what the linear response of a system is as a quantification of sensitivity alongside previous work here we use the transfer operator formalism to study the response and sensitivity of a dynamical system undergoing perturbations by projecting the transfer operator onto a suitable finite dimensional vector space one is able to obtain matrix representations which determine finite markov processes further using perturbation theory for markov matrices it is possible to determine the linear and nonlinear response of the system given a prescribed forcing here we suggest a methodology which puts the scope on the evolution law of densities the liouville fokker planck equation allowing to effectively calculate the sensitivity and response of two representative dynamical systems

torus invariant prime ideals in quantum matrices totally nonnegative cells and symplectic leaves the algebra of quantum matrices of a given size supports a rational torus action by automorphisms it follows from work of letzter and the first named author that to understand the prime and primitive spectra of this algebra the first step is to understand the prime ideals that are invariant under the torus action in this paper we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size as a corollary we obtain explicit generating sets of quantum minors for the torus invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over mathbb q

schubert complexes and degeneracy loci the classical thom porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of schur polynomials a proof of this formula was given by pragacz by expressing this alternating sum as the euler characteristic of a schur complex which gives an explanation for the signs fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of schubert polynomials building on the schubert functors of kraand pragacz we introduce schubert complexes and show that fulton s alternating sum can be realized as the euler characteristic of this complex thereby providing a conceptual proof for why an alternating sum appears

totally nonnegative cells and matrix poisson varieties we describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices that is the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by a postnikov in order to do this we relate the totally nonnegative cells to torus orbits of symplectic leaves of the poisson varieties of complex matrices in particular we describe the minors that vanish on a torus orbit of symplectic leaves we prove that such families of minors are exactly the admissible families and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices

the geometric minimum action method a least action principle on the space of curves freidlin wentzell theory of large deviations for the description of the effect of small random perturbations on dynamical systems is exploited as a numerical tool specifically a numerical algorithm is proposed to compute the quasi potential in the theory which is the key object to quantify the dynamics on long time scales when the effect of the noise becomes ubiquitous the equilibrium distribution of the system the pathways of transition between metastable states and their rate etc can all be expressed in terms of the quasi potential we propose an algorithm to compute these quantities called the geometric minimum action method gmam which is a blend of the original minimum action method mam and the string method it is based on a reformulation of the large deviations action functional on the space of curves that allows one to easily perform the double minimization of the original action required to compute the quasi potential the theoretical background of the gmam in the context of large deviations theory is discussed in detail as well as the algorithmic aspects of the method the gmam is then illustrated on several examples a finite dimensional system displaying bistability and modeled by a nongradient stochastic ordinary differential equation an infinite dimensional analogue of this system modeled by a stochastic partial differential equation and an example of a bistable genetic switch modeled by a markov jump process

an improved version of marti s method for solving ill posed linear integral equations we propose an algorithm for solving linear integral equations of the first kind that can be viewed as a variant of marti s method as opposed to that method our algorithm leads to optimal convergence rates also with noisy data 1 introduction throughout this paper let x and y be real hilbert spaces t x y a bounded linear operator with nonclosed range then the problem of determining the best approximate solution of 1 1 tx y is ill posed the best approximate solution exists only for y e d tt r t r t which we assume from now on and depends discontinuously on the right hand side here tt is the moore penrose inverse of t see 12 the best approximate solution is defined as the element of minimal norm that minimizes the residual tx yll and can be written as tty a prominent example for the ill posed

positivity of quiver coefficients through thom polynomials abstract we use the thom polynomial theory developed by feher and rimanyi to prove the component formula for quiver varieties conjectured by knutson miller and shimozono this formula expresses the cohomology class of a quiver variety as a sum of products of schubert polynomials indexed by minimal lace diagrams and implies that the quiver coefficients of buch and fulton are non negative we also apply our methods to give a new proof of the component formula from the grobner degeneration of quiver varieties and to give generating moves for the kms factorizations that form the index set in k theoretic versions of the component formula

combinatorics of fulton s essential set 1 introduction an n by n permutation matrix can be represented by an nby n array of squares with one dot in each row and column and all other squares empty the diagram of a permutation matrix defined in 1800 by rothe is obtained by shading every row from the dot and eastwards and shading every column from the dot and southwards the essential set is the set of southeast corners of the connected components of the diagram the essential set together with a rank function was introduced by fulton 7 in a pioneering paper from 1992 to study the irreducible loci in spaces of matrices although we are not going to pursue this geometric issue much but rather concentrate on the essential set per se we need a short recollection of the background fulton studies varieties given by ideals of minors subject to certain rank conditions in a generic matrix one concern is given a prescribed rank function r i j to determine if there exist matrices such that the rank of the upper left i by j submatrices is r i j for every position i j he observes that if such matrices exist then there is in particular some permutation matrix with this property so it is enough to consider permutation matrices more generally he is interested in prescribing the ranks only for a few of the submatrices such that all the other ranks follow from these in other words to find a subset of the upper left submatrices of a permutation matrix such that the permutation is uniquely determined by their ranks fulton shows that the essential set of a permutation matrix is such a set that is the permutation is determined by its essential set and the ranks of the corresponding upper left submatrices in section 2 we present more detailed definitions and the relevant results of fulton our first result in section 3 is an elementary algorithm that retrieves a permutation from its ranked essential set the algorithm is flexible in that the input set may be bigger and include any other squares as well as be smaller as long as it contains a certain core of the essential set by this flexibility the algorithm ought to be useful as a practical tool in general the core is much smaller than the essential set so this is a significant strengthening fulton used the essential sets mainly for vexillary 2143 avoiding permutations he gave a characterization of the ranked essential set for vexillary permutations which may be expressed by three simple conditions and he pointed out that it might be useful to have an analogous characterization of the ranked sets of squares that arise as ranked essential sets of all permutations as an application of the algorithm we are able to state such a characterization which keeps

finite element approach to clustering of multidimensional time series we present a new approach to clustering of time series based on a minimization of the averaged clustering functional the proposed functional describes the mean distance between observation data and its representation in terms of mathbf k abstract models of a certain predefined class not necessarily given by some probability distribution for a fixed time series x t this functional depends on mathbf k sets of model parameters theta theta 1 dots theta mathbf k and mathbf k functions of cluster affiliations gamma gamma 1 t dots gamma mathbf k t characterizing the affiliation of any element x t of the analyzed time series to one of the mathbf k clusters defined by the considered model parameters we demonstrate that for a fixed set of model parameters theta the appropriate tykhonov type regularization of this functional with some regularization factor epsilon 2 results in a minimization problem similar to a variational problem usually associated with one dimensional nonhomogeneous partial differential equations this analogy allows us to apply the finite element framework to the problem of time series analysis and to propose a numerical scheme for time series clustering we investigate the conditions under which the proposed scheme allows a monotone improvement of the initial parameter guess with respect to the minimization of the discretized version of the regularized functional we also discuss the interpretation of the regularization factor in the markovian case and show its connection to metastability and exit times the computational performance of the resulting method is investigated numerically on multidimensional test data and is applied to the analysis of multidimensional historical stock market data

the reduction of complex dynamical systems using principal interaction patterns abstract a method of constructing low dimensional nonlinear models capturing the main features of complex dynamical systems with many degrees of freedom is described the system is projected onto a linear subspace spanned by only a few characteristic spatial structures called principal interaction patterns pips the expansion coefficients are assumed to be governed by a nonlinear dynamical system the optimal low dimensional model is determined by identifying spatial modes and interaction coefficients describing their time evolution simultaneously according to a nonlinear variational principle the algorithm is applied to a two dimensional geophysical fluid system on the sphere the models based on principal interaction patterns are compared to models using empirical orthogonal functions eofs as basis functions a pip model using 12 patterns is capable of capturing the long term behaviour of the complete system monitored by second order statistics while in the case of eofs 17 modes are necessary

statistical analysis of the first passage path ensemble of jump processes the transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years in this paper we study the first passage path ensemble of both discrete time and continuous time jump processes on a finite state space the main approach is to divide each first passage path into nonreactive and reactive segments and to study them separately the analysis can be applied to jump processes which are non ergodic as well as continuous time jump processes where the waiting time distributions are non exponential in the particular case that the jump processes are both markovian and ergodic our analysis elucidates the relations between the study of the first passage paths and the study of the transition paths in transition path theory we provide algorithms to numerically compute statistics of the first passage path ensemble the computational complexity of these algorithms scales with the complexity of solving a linear system for which efficient methods are available several examples demonstrate the wide applicability of the derived results across research areas

segre classes and kempf laksov formula in algebraic cobordism in this paper we extend to algebraic cobordism the classical damon kempf laksov formula which expresses the chow ring schubert classes of grassmann bundles as schur determinants in chern classes the basic building block of our closed formula which is written as a sum of determinants is represented by a generalisation of segre classes that we introduce and describe

optimal path to epigenetic switching we use large deviation methods to calculate rates of noise induced transitions between states in multistable genetic networks we analyze a synthetic biochemical circuit the toggle switch and compare the results to those obtained from a numerical solution of the master equation

escape problem for irreversible systems the problem of noise induced escape from a metastable state arises in physics chemistry biology systems engineering and other areas the problem is well understood when the underlying dynamics of the system obey detailed balance when this assumption fails many of the results of classical transition rate theory no longer apply and no general method exists for computing the weak noise asymptotic behavior of fundamental quantities such as the mean escape time in this paper we present a general technique for analyzing the weak noise limit of a wide range of stochastically perturbed continuous time nonlinear dynamical systems we simplify the original problem which involves solving a partial differential equation into one in which only ordinary differential equations need be solved this allows us to resolve some old issues for the case when detailed balance holds when it does not hold we show how the formula for the asymptotic behavior of the mean escape time depends on the dynamics of the system along the most probable escape path we also present results on short time behavior and discuss the possibility of ital focusing along the escape path

a dirichlet process gaussian state machine model for change detection in transient processes abstractthe ability to detect incipient and critical changes in real world process esessential for system integrity assurance is currently impeded by the mismatch between the key assumption of stationarity underlying most change detection methods and the nonlinear and nonstationary transient dynamics of most real world processes the current approaches are slow or outright unable to detect qualitative changes in the behaviors that lead to anomalies we present a dirichlet process gaussian state machine dpgsm model to represent dynamic intermittency which is one of the most ubiquitous real world transient behaviors the dpgsm model treats a signal as a random walk among a dirichlet process mixture of gaussian clusters hypothesis tests and a numerical scheme based on this nonparametric representation were developed to detect subtle changes in the transient intermittent dynamics experimental investigations suggest that the dpgsm approach can consistently detect incipient critical changes in intermi

statistical inference for discretely observed markov jump processes summary likelihood inference for discretely observed markov jump processes with finite state space is investigated the existence and uniqueness of the maximum likelihood estimator of the intensity matrix are investigated this topic is closely related to the imbedding problem for markov chains it is demonstrated that the maximum likelihood estimator can be found either by the em algorithm or by a markov chain monte carlo procedure when the maximum likelihood estimator does not exist an estimator can be obtained by using a penalized likelihood function or by the markov chain monte carlo procedure with a suitable prior the methodology and its implementation are illustrated by examples and simulation studies

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

k theoretic chern class formulas for vexillary degeneracy loci abstract using raising operators and geometric arguments we establish formulas for the k theory classes of degeneracy loci in classical types we also find new determinantal and pfaffian expressions for classical cases considered by giambelli the loci where a generic matrix drops rank and where a generic symmetric or skew symmetric matrix drops rank in an appendix we construct a k theoretic euler class for even rank vector bundles with quadratic form refining the chow theoretic class introduced by edidin and graham we also establish a relation between top chern classes of maximal isotropic subbundles which is used in proving the type d degeneracy locus formulas

a scheme related to the brauer loop model abstract we introduce the brauer loop scheme e m m n c m m 0 where is a certain degeneration of the ordinary matrix product its components of top dimension n 2 2 correspond to involutions s n having one or no fixed points in the case n even this scheme contains the upper upper scheme from a knutson some schemes related to the commuting variety j algebraic geom in press math ag 0306275 as a union of n 2 of its components one of those is a degeneration of the commuting variety of pairs of commuting matrices the brauer loop model is an integrable stochastic process studied in j de gier b nienhuis brauer loops and the commuting variety j stat mech 2005 p01006 math ag 0410392 based on earlier related work in m j martins b nienhuis r rietman an intersecting loop model as a solvable super spin chain phys rev lett 81 1998 504 507 cond mat 9709051 and some of the entries of its perron frobenius eigenvector were observed conjecturally to equal the degrees of the components of the upper upper scheme our proof of this equality follows the program outlined in p di francesco p zinn justin inhomogeneous model of crossing loops and multidegrees of some algebraic varieties math ph 0412031 in that paper the entries of the perron frobenius eigenvector were generalized from numbers to polynomials which allowed them to be calculated inductively using divided difference operators we relate these polynomials to the multi degrees of the components of the brauer loop scheme defined using an evident torus action on e as a consequence we obtain a formula for the degree of the commuting variety previously calculated up to 4 4 matrices

every abelian l group is ultrasimplicial abstract a partially ordered abelian group g is said to be ultrasimplicial if for every finite set p of positive elements of g there is a finite set b of positive elements which are linearly independent in the z module g and such that p belongs to the monoid generated by b in this paper we prove the result stated in the title

estimation for general birth death processes birth death processes bdps are continuous time markov chains that track the number of particles in a system over time while widely used in population biology genetics and ecology statistical inference of the instantaneous particle birth and death rates remains largely limited to restrictive linear bdps in which per particle birth and death rates are constant researchers often observe the number of particles at discrete times necessitating data augmentation procedures such as expectation maximization em to find maximum likelihood estimates mles for bdps on finite state spaces there are powerful matrix methods for computing the conditional expectations needed for the e step of the em algorithm for bdps on infinite state spaces closed form solutions for the e step are available for some linear models but most previous work has resorted to time consuming simulation remarkably we show that the e step conditional expectations can be expressed as convolutions of computable transition probabi

direct calculation of the information matrix via the em algorithm summary a simple explicit formula is provided for the matrix of second derivatives of the observed data log likelihood in terms of derivatives of the criterion function the conditional expectation of the complete data log likelihood given the observed data invoked by the em algorithm

an exact gibbs sampler for the markov modulated poisson process a markov modulated poisson process mmpp is a poisson process whose intensity varies according to a markov process we present a novel technique for simulating from the exact distribution of a continuous time markov chain over an interval given the start and end states and the infinitesimal generator and use this to create a gibbs sampler which samples from the exact distribution of the hidden markov chain in an mmpp we apply the gibbs sampler to modelling the occurrence of a rare dna motif the chi site and to inferring regions of the genome with evidence of high or low intensities for occurrences of this site

a numerical scheme for optimal transition paths of stochastic chemical kinetic systems we present a new framework for finding the optimal transition paths of metastable stochastic chemical kinetic systems with large system size the optimal transition paths are identified to be the most probable paths according to the large deviation theory of stochastic processes dynamical equations for the optimal transition paths are derived using the variational principle a modified minimum action method mam is proposed as a numerical scheme to solve the optimal transition paths applications to gene regulatory networks such as the toggle switch model and the lactose operon model in escherichia coli are presented as numerical examples

fitting timeseries by continuous time markov chains a quadratic programming approach construction of stochastic models that describe the effective dynamics of observables of interest is an useful instrument in various fields of application such as physics climate science and finance we present a new technique for the construction of such models from the timeseries of an observable we construct a discrete in time markov chain and calculate the eigenspectrum of its transition probability or stochastic matrix as a next step we aim to find the generator of a continuous time markov chain whose eigenspectrum resembles the observed eigenspectrum as closely as possible using an appropriate norm the generator is found by solving a minimization problem the norm is chosen such that the object function is quadratic and convex so that the minimization problem can be solved using quadratic programming techniques the technique is illustrated on various toy problems as well as on datasets stemming from simulations of molecular dynamics and of atmospheric flows

direct calculation of the information matrix via the em a simple explicit formula is provided for the matrix of second derivatives of the observed data log likelihood in terms of derivatives of the criterion function conditional expectation of the complete data log likelihood given the observed data invoked by the em algorithm

lattice ordered abelian groups and schauder bases of unimodular fans baker beynon duality theory yields a concrete representation of any finitely generated projective abelian lattice ordered group g in terms of piecewise linear homogeneous functions with integer coefficients defined over the support e of a fan a unimodular fan a over s determines a schauder basis of g its elements are the minimal positive free generators of the pointwise ordered group of a linear support functions conversely a schauder basis h of g determines a unimodular fan over its maximal cones are the domains of linearity of the elements of h the main purpose of this paper is to give various representation free characterisations of schauder bases the latter jointly with the de concini procesi starring technique will be used to give novel characterisations of finitely generated projective abelian lattice ordered groups for instance g is finitely generated projective iff it can be presented by a purely lattice theoretical word

towards a theory of transition paths we construct a statistical theory of reactive trajectories between two pre specified sets a and b i e the portionsof the path of a markov process during which the path makes a transition from a to b this problem is relevant e g in the context of metastability in which case the two sets a and b are metastable sets though the formalism we propose is independent of any such assumptions on a and b we show that various probability distributions on the reactive trajectories can be expressed in terms of the equilibrium distribution of the process and the so called committor functions which give the probability that the process reaches first b before reaching a either backward or forward in time using these objects we obtain i the distribution of reactive trajectories which gives the proportion of time reactive trajectories spend in sets outside of a and b ii the hitting point distribution of the reactive trajectories on a surface which measures where the reactive trajectories hit the surface when they cross it iii the last hitting point distribution of the reactive trajectories on the surface iv the probability current of reactive trajectories the integral of which on a surface gives the net average flux of reactive trajectories across this surface v the average frequency of reactive trajectories which gives the average number of transitions between a and b per unit of time and vi the traffic distribution of reactive trajectories which gives some information about the regions the reactive trajectories visit regardless of the time they spend in these regions

a directed graph structure of alternating sign matrices abstract we introduce a new directed graph structure into the set of alternating sign matrices this includes bruhat graph bruhat order of the symmetric groups as a subgraph subposet drake gerrish skandera 2004 2006 6 7 gave characterizations of bruhat order in terms of total nonnegativity tnn and subtraction free laurent sfl expressions for permutation monomials with our directed graph we extend their idea in two ways first from permutations to alternating sign matrices second q analogs which we name q tnn and q sfl properties as a by product we obtain a new kind of permutation statistic the signed bigrassmannian statistics using dodgson s condensation on determinants

when is a schubert variety gorenstein abstract a normal variety is gorenstein if it is cohen macaulay and its canonical sheaf is a line bundle this property is a weakening of smoothness which implies that a variety behaves like a smooth one for various algebro geometric purposes we give a combinatorial characterization of which schubert varieties are gorenstein using in part a new generalization of pattern avoidance we also give an explicit description as a line bundle of the canonical sheaf of a gorenstein schubert variety

a grobner basis for kazhdan lusztig ideals it kazhdan lusztig ideals a family of generalized determinantal ideals investigated in woo yong 08 provide an explicit choice of coordinates and equations encoding a neighborhood of a torus fixed point of a schubert variety on a type a flag variety our main result is a gr o bner basis for these ideals this provides a single geometric setting to transparently explain the naturality of pipe dreams on the it rothe diagram of a permutation and their appearance in begin itemize item combinatorial formulas fomin kirillov 94 for schubert and grothendieck polynomialsof lascoux sch u tzenberger 82 item the equivariant k theory specialization formula of buch rim a nyi 04 and item a positive combinatorial formula for multiplicities of schubert varieties in good cases including those for which the associated kazhdan lusztig ideal is homogeneous under the standard grading end itemize our results generalize with alternate proofs knutson miller 05 s gr o bner basis theorem for schubert determinantal ideals and their geometric interpretation of the monomial positivity of schubert polynomials we also complement recent work of knutson 08 09 on degenerations of kazhdan lusztig varieties in general lie type as well as work of goldin 01 on equivariant localization and of lakshmibai weyman 90 rosenthal zelevinsky 01 and krattenthaler 01 on grassmannian multiplicity formulas

classes of ultrasimplicial lattice ordered abelian groups abstract a lattice ordered abelian group is called ultrasimplicial iff every finite set of positive elements belongs to the monoid generated by some finite set of positive z independent elements this property originates from elliott s classification of af c algebras using fans and their desingularizations it is proved that the ultrasimplicial property holds for every n generated archimedean l group whose maximal l ideals of rank n are dense as a corollary we obtain simpler proofs of results respectively by elliott and by the present author stating that totally ordered abelian groups as well as free l groups are ultrasimplicial

generator estimation of markov jump processes based on incomplete observations nonequidistant in time markov jump processes can be used to model the effective dynamics of observables in applications ranging from molecular dynamics to finance in this paper we present a different method which allows the inverse modeling of markov jump processes based on incomplete observations in time we consider the case of a given time series of the discretely observed jump process we show how to compute efficiently the maximum likelihood estimator of its infinitesimal generator and demonstrate in detail that the method allows us to handle observations nonequidistant in time the method is based on the work of and bladt and sorensen j r stat soc ser b stat methodol 67 395 2005 but scales much more favorably than it with the length of the time series and the dimension and size of the state space of the jump process we illustrate its performance on a toy problem as well as on data arising from simulations of biochemical kinetics of a genetic toggle switch

some degenerations of kazhdan lusztig ideals and multiplicities of schubert varieties abstract we study hilbert samuel multiplicity for points of schubert varieties in the complete flag variety by grobner degenerations of the kazhdan lusztig ideal in the covexillary case we give a manifestly positive combinatorial rule for multiplicity by establishing with a grobner basis a reduced limit whose stanley reisner simplicial complex is homeomorphic to a shellable ball or sphere we show that multiplicity counts the number of facets of this complex we also obtain a formula for the hilbert series of the local ring in particular our work gives a multiplicity rule for grassmannian schubert varieties providing alternative statements and proofs to formulae of lakshmibai and weyman 1990 26 rosenthal and zelevinsky 2001 37 krattenthaler 2001 22 kodiyalam and raghavan 2003 21 kreiman and lakshmibai 2004 24 ikeda and naruse 2009 13 and woo and yong 2009 40 we suggest extensions of our methodology to the general case

on quantum cohomology rings of partial flag varieties 0 introduction the main goal of this paper is to give a unified description for the structure of the small quantum cohomology rings for all projective homogeneous spaces sln c p where p is a parabolic subgroup the quantum cohomology ring of a smooth projective variety or more generally of a symplectic manifoldx has been introduced by string theorists see va and w1 roughly speaking it is a deformation of the usual cohomology ring with parameter space given byh x the multiplicative structure of quantum cohomology encodes the enumerative geometry of rational curves on x in the past few years the highly nontrivial task of giving a rigorous mathematical treatment of quantum cohomology has been accomplished both in the realm of algebraic and symplectic geometry in various degrees of generality this can be found in beh behm km lit1 lit2 mcs and rt as well as in the surveys fp and t if one restricts the parameter space to h 1 1 x one gets the small quantum cohomology ring this ring in the case of partial flag varieties is the object of the present paper in order to state our main results we first describe briefly the classical side of the story we interpret the homogeneous space f sln c p as the complex projective variety parametrizing flags of quotients of c of given ranks say nk n1 by a classical result of ehresmann e the integral cohomology of f can be described geometrically as the free abelian group generated by the schubert classes these are the poincare duals of fundamental classes of certain subvarieties w f one for each element of the subset s s n1 nk of the symmetric group sn consisting of permutations w with descents in n1 nk a description of the multiplicative structure is provided by yet another classical theorem due to borel bor which gives a presentation forh f z specifically let 1 1 1 n1 2 1 2 n2 n1 k 1 1 k 1 n nk be n independent variables define an to be the block diagonal matrix diag d1 d2 dk 1 where

the nash blow up of a cominuscule schubert variety abstract we compute the nash blow up of a cominuscule schubert variety in particular we show that the nash blow up is algebraically isomorphic to another schubert variety of the same lie type as a consequence we give a new characterization of the smooth locus and for grassmannian schubert varieties determine when the nash blow up is a resolution of singularities we also study the induced torus action on the nash blow up and give a bijection between its torus fixed points and peterson translates on the schubert variety

generator estimation of markov jump processes estimating the generator of a continuous time markov jump process based on incomplete data is a problem which arises in various applications ranging from machine learning to molecular dynamics several methods have been devised for this purpose a quadratic programming approach cf d t crommelin e vanden eijnden fitting timeseries by continuous time markov chains a quadratic programming approach j comp phys 217 2006 782 805 a resolvent method cf t muller modellierung von proteinevolution phd thesis heidelberg 2001 and various implementations of an expectation maximization algorithm s asmussen o nerman m olsson fitting phase type distributions via the em algorithm scand j stat 23 1996 419 441 i holmes g m rubin an expectation maximization algorithm for training hidden substitution models j mol biol 317 2002 753 764 u nodelman c r shelton d koller expectation maximization and complex duration distributions for continuous time bayesian networks in proceedings of the twenty first conference on uncertainty in ai uai 2005 pp 421 430 m bladt m sorensen statistical inference for discretely observed markov jump processes j r statist soc b 67 2005 395 410 some of these methods however seem to be known only in a particular research community and have later been reinvented in a different context the purpose of this paper is to compile a catalogue of existing approaches to compare the strengths and weaknesses and to test their performance in a series of numerical examples these examples include carefully chosen model problems and an application to a time series from molecular dynamics

a score test on measurement errors in rating transition times abstract we model credit rating histories as continuous time discrete state markov processes infrequent monitoring of the debtors solvency will result in erroneous observations of the rating transition times and consequently in biased parameter estimates we develop a score test against such measurement errors in the transition data that is independent of the error distribution we derive the asymptotic 2 distribution for the test statistic under the null by stochastic limit theory the test is applied to an international corporate portfolio while accounting for economic and debtor specific covariates the test indicates that measurement errors in the transition times are a real problem in practice

an ensemble kalman filter for statistical estimation of physics constrained nonlinear regression models a central issue in contemporary science is the development of nonlinear data driven statistical dynamical models for time series of noisy partial observations from nature or a complex model it has been established recently that ad hoc quadratic multi level regression models can have finite time blow up of statistical solutions and or pathological behavior of their invariant measure recently a new class of physics constrained nonlinear regression models were developed to ameliorate this pathological behavior here a new finite ensemble kalman filtering algorithm is developed for estimating the state the linear and nonlinear model coefficients the model and the observation noise covariances from available partial noisy observations of the state several stringent tests and applications of the method are developed here in the most complex application the perfect model has 57 degrees of freedom involving a zonal east west jet two topographic rossby waves and 54 nonlinearly interacting rossby waves the perfect model has significant non gaussian statistics in the zonal jet with blocked and unblocked regimes and a non gaussian skewed distribution due to interaction with the other 56 modes we only observe the zonal jet contaminated by noise and apply the ensemble filter algorithm for estimation numerically we find that a three dimensional nonlinear stochastic model with one level of memory mimics the statistical effect of the other 56 modes on the zonal jet in an accurate fashion including the skew non gaussian distribution and autocorrelation decay on the other hand a similar stochastic model with zero memory levels fails to capture the crucial non gaussian behavior of the zonal jet from the perfect 57 mode model