assessing the order of dependence for partially exchangeable binary data abstract the problem we consider is how to assess the order of serial dependence within partially exchangeable binary sequences we obtain exact conditional tests comparing any two orders by finding the conditional distribution of data given certain transition counts these tests are facilitated with a new monte carlo scheme asymptotic tests are also discussed in particular we show that the likelihood ratio tests have an asymptotic chi squared distribution thus generalizing the results of billingsley for the particular case of markov chains we apply these methods to several datasets and perform a simulation to study their properties

matroid configurations and symbolic powers of their ideals star configurations are certain unions of linear subspaces of projective space that have been studied extensively we develop a framework for studying a substantial gen eralization which we call matroid configurations whose ideals generalize stanley reisner ideals of matroids such a matroid configuration is a union of complete intersections of a fixed codimension relating these to the stanley reisner ideals of matroids and using methods of liaison theory allows us in particular to describe the hilbert function and minimal generators of the ideal of what we call a hypersurface configuration we also establish that the symbolic powers of the ideal of any matroid configuration are cohen macaulay as applications we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves we also consider waldschmidt constants and resurgences in particular we determine the resurgence of any star configuration and many hypersurface configurations previously the only non trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points finally we point out a connection to secant varieties of varieties of reducible forms

generalized monte carlo significance tests summary simple monte carlo significance testing has many applications particularly in the preliminary analysis of spatial data the method requires the value of the test statistic to be ranked among a random sample of values generated according to the null hypothesis however there are situations in which a sample of values can only be conveniently generated using a markov chain initiated by the observed data so that independence is violated this paper describes two methods that overcome the problem of dependence and allow exact tests to be carried out the methods are applied to the rasch model to the finite lattice ising model and to the testing of association between spatial processes power is discussed in a simple case

containment problem for points on a reducible conic in p2 abstract given an ideal i in a noetherian ring one can ask the containment question for which m and r is the symbolic power i m contained in the ordinary power i r c bocci and b harbourne study the containment question in a geometric setting where the ideal i is in a polynomial ring over a field like them we will consider special geometric constructs in particular we obtain a complete solution in two extreme cases of ideals of points on a pair of lines in p 2 in one case the number of points on each line is the same while in the other all the points but one are on one of the lines

star configurations on generic hypersurfaces let f be a homogeneous polynomial in s c x0 xn our goal is to understand a particular polynomial decomposition of f geometrically we wish to determine when the hypersurface defined by f in pn contains a star configuration to solve this problem we use techniques from commutative algebra and algebraic geometry to reduce our question to computing the rank of a matrix

betti numbers of symmetric shifted ideals abstract we introduce a new class of monomial ideals which we call symmetric shifted ideals symmetric shifted ideals are fixed by the natural action of the symmetric group and within the class of monomial ideals fixed by this action they can be considered as an analogue of stable monomial ideals within the class of monomial ideals we show that a symmetric shifted ideal has linear quotients and compute its equivariant graded betti numbers as an application of this result we obtain several consequences for graded betti numbers of symbolic powers of defining ideals of star configurations

finite generation of symmetric ideals let be a commutative noetherian ring and let be the polynomial ring in an infinite collection of indeterminates over let be the group of permutations of the group acts on in a natural way and this in turn gives the structure of a left module over the group ring we prove that all ideals of invariant under the action of are finitely generated as modules the proof involves introducing a certain well quasi ordering on monomials and developing a theory of grobner bases and reduction in this setting we also consider the concept of an invariant chain of ideals for finite dimensional polynomial rings and relate it to the finite generation result mentioned above finally a motivating question from chemistry is presented with the above framework providing a suitable context in which to study it

some formulas for invariant phases of unitary matrices by jarlskog we describe a calculation of jarlskog s determinant in the case of n 4 in detail next we investigate some formulas for invariant phases of unitary matrices and derive some explicit relations for them

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

on the componentwise linearity and the minimal free resolution of a tetrahedral curve abstract a tetrahedral curve is an unmixed usually nonreduced one dimensional subscheme of projective 3 space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines the second and third authors have shown that these curves have very nice combinatorial properties and they have made a careful study of the even liaison classes of these curves we build on this work by showing that they are almost always componentwise linear i e their homogeneous ideals have the property that for any d the degree d component of the ideal generates a new ideal whose minimal free resolution is linear the one type of exception is clearly spelled out and studied as well the main technique is a careful study of the way that basic double linkage behaves on tetrahedral curves and the connection to the tetrahedral curves that are minimal in their even liaison classes with this preparation we also describe the minimal free resolution of a tetrahedral curve and in particular we show that in any fixed even liaison class there are only finitely many tetrahedral curves with linear resolution finally we begin the study of the generic initial ideal gin of a tetrahedral curve we produce the gin for arithmetically cohen macaulay tetrahedral curves and for minimal arithmetically buchsbaum tetrahedral curves and we show how to obtain it for any nonminimal tetrahedral curve in terms of the gin of the minimal curve in that even liaison class

exact goodness of fit tests for markov chains goodness of fit tests are useful in assessing whether a statistical model is consistent with available data however the usual asymptotics often fail either because of the paucity of the data or because a nonstandard test statistic is of interest in this article we describe exact goodness of fit tests for first and higher order markov chains with particular attention given to time reversible ones the tests are obtained by conditioning on the sufficient statistics for the transition probabilities and are implemented by simple monte carlo sampling or by markov chain monte carlo they apply both to single and to multiple sequences and allow a free choice of test statistic three examples are given the first concerns multiple sequences of dry and wet january days for the years 1948 1983 at snoqualmie falls washington state and suggests that standard analysis may be misleading the second one is for a four state dna sequence and lends support to the original conclusion that a second order markov chain provides an adequate fit to the data the last one is six state atomistic data arising in molecular conformational dynamics simulation of solvated alanine dipeptide and points to strong evidence against a first order reversible markov chain at 6 picosecond time steps

on the first infinitesimal neighborhood of a linear configuration of points in p2 abstract we consider the following open questions fix a hilbert function h that occurs for a reduced zero dimensional subscheme of p 2 among all subschemes x with hilbert function h what are the possible hilbert functions and graded betti numbers for the first infinitesimal neighborhood z of x i e the double point scheme supported on x is there a minimum h min and maximum h max such function the numerical information encoded in h translates to a type vector which allows us to find unions of points on lines called linear configurations with hilbert function h we give necessary and sufficient conditions for the hilbert function and graded betti numbers of the first infinitesimal neighborhoods of all such linear configurations to be the same even for those h for which the hilbert functions or graded betti numbers of the resulting double point schemes are not uniquely determined we give one depending only on h that does occur we prove the existence of h max in general and discuss h min our methods include liaison techniques

bayesian analysis of variable order reversible markov chains we define a conjugate prior for the reversible markov chain of order r the prior arises from a partially exchangeable reinforced random walk in the same way that the beta distribution arises from the exchangeable polya urn an extension to variable order markov chains is also derived we show the utility of this prior in testing the order and estimating the parameters of a reversible markov model

big rational surfaces we prove that the cox ring of a smooth rational surface with big anticanonical class is finitely generated we classify surfaces of this type that are blow ups of mathbb p 2 at distinct points lying on a possibly reducible cubic

dna viewed as an out of equilibrium structure the complexity of the primary structure of human dna is explored using methods from nonequilibrium statistical mechanics dynamical systems theory and information theory a collection of statistical analyses is performed on the dna data and the results are compared with sequences derived from different stochastic processes the use of 2 tests shows that dna can not be described as a low order markov chain of order up to r 6 although detailed balance seems to hold at the level of a binary alphabet it fails when all four base pairs are considered suggesting spatial asymmetry and irreversibility furthermore the block entropy does not increase linearly with the block size reflecting the long range nature of the correlations in the human genomic sequences to probe locally the spatial structure of the chain we study the exit distances from a specific symbol the distribution of recurrence distances and the hurst exponent all of which show power law tails and long range characteristics these results suggest that human dna can be viewed as a nonequilibrium structure maintained in its state through interactions with a constantly changing environment based solely on the exit distance distribution accounting for the nonequilibrium statistics and using the monte carlo rejection sampling method we construct a model dna sequence this method allows us to keep both long and short range statistical characteristics of the native dna data the model sequence presents the same characteristic exponents as the natural dna but fails to capture spatial correlations and point to point details 2014 american physical society

rephasing invariants of the cabibbo kobayashi maskawa matrix the paper is motivated by the importance of the rephasing invariance of the ckm cabibbo kobayashi maskawa matrix observables these observables appear in the discussion of the cp violation in the standard model jarlskog invariant and also in the renormalization group equations for the quark yukawa couplings our discussion is based on the general phase invariant monomials built out of the ckm matrix elements and their conjugates we show that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 ckm matrix elements and 12 are a product of 6 ckm matrix elements in the main theorem we show that a general rephasing invariant monomial can be expressed as a product of at most five factors four of them are fundamental phase invariant monomials and the fifth factor consists of powers of squares of absolute values of the ckm matrix elements we also show that the imaginary part of any rephasing invariant monomial is proportional to the jarlskog s invariant j or is 0

exchangeability in multivariate markov chain models time homogeneous markov chain models with state space 0 1 k are useful in analysis of binary follow up data on k individuals that interact the number of parameters increases exponentially with k so more restrictive models are imperative for statistical inference the hypothesis that the matrix of transition probabilities is invariant under permutation of individuals is discussed it is shown that if individuals are exchangeable then the process counting the number of individuals occupying a given state is a markov chain this reduction of data is sufficient if either at most a single individual may change state between two consecutive time points or if a state is absorbing similar results are obtained for exchangeability within two subgroups inference in the multivariate process reduces to a univariate problem if individuals are independent given the group s previous response it is shown how conditional independence could be tested assuming exchangeability the different hypotheses re examined in an analysis of the occurrence of bacteria in milk samples of danish dairy cattle

the effect of simulation order on level accuracy and power of monte carlo tests we discuss the effect of simulation order on level accuracy and power of monte carlo tests in a very general setting both parametric problems with or without nuisance parameters and nonparametric problems are treated by a single unifying argument it is shown that if the level of a monte carlo test is known only nominally not precisely then the level error of a monte carlo test is an order of magnitude less than that of the corresponding asymptotic test this result is available whenever the test statistic is asymptotically pivotal even if the number of simulations is held fixed as the sample size n increases it implies that monte carlo methods are a real alternative to asymptotic methods we also show that even if the number of simulations is held fixed a monte carlo test is able to distinguish between the null hypothesis and alternative hypotheses distant n 12 from the null

bayesian analysis for reversible markov chains we introduce a natural conjugate prior for the transition matrix of a reversible markov chain this allows estimation and testing the prior arises from random walk with reinforcement in the same way the dirichlet prior arises from polya s urn we give closed form normalizing constants a simple method of simulation from the posterior and a characterization along the lines of w e johnson s characterization of the dirichlet prior

star configurations in pn abstract star configurations are certain unions of linear subspaces of projective space they have appeared in several different contexts the study of extremal hilbert functions for fat point schemes in the plane the study of secant varieties of some classical algebraic varieties the study of the resurgence of projective schemes in this paper we study some algebraic properties of the ideals defining star configurations including getting partial results about hilbert functions generators and minimal free resolutions of the ideals and their symbolic powers we also show that their symbolic powers define arithmetically cohen macaulay subschemes and we obtain results about the primary decompositions of the powers of the ideals as an application we compute the resurgence for the ideal of the codimension n 1 star configuration in p n in the monomial case i e when the number of hyperplanes is n 1

bayesian nonparametric modeling of higher order markov chains we consider the problem of flexible modeling of higher order markov chains when an upper bound on the order of the chain is known but the true order and nature of the serial dependence are unknown we propose bayesian nonparametric methodology based on conditional tensor factorizations which can characterize any transition probability with a specified maximal order the methodology selects the important lags and captures higher order interactions among the lags while also facilitating calculation of bayes factors for a variety of hypotheses of interest we design efficient markov chain monte carlo algorithms for posterior computation allowing for uncertainty in the set of important lags to be included and in the nature and order of the serial dependence the methods are illustrated using simulation experiments and real world applications supplementary materials for this article are available online

time reversibility of stationary regular finite state markov chains we propose an alternate parameterization of stationary regular finite state markov chains and a decomposition of the parameter into time reversible and time irreversible parts we demonstrate some useful properties of the decomposition and propose an index for a certain type of time irreversibility applicable to chains whose states are naturally ordered two empirical examples illustrate the use of the proposed parameter decomposition and index one on gasoline price mark ups involves observed states the other on u s investment growth features latent states

finite grobner bases in infinite dimensional polynomial rings and applications abstract we introduce the theory of monoidal grobner bases a concept which generalizes the familiar notion in a polynomial ring and allows for a description of grobner bases of ideals that are stable under the action of a monoid the main motivation for developing this theory is to prove finiteness results in commutative algebra and applications a basic theorem of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry using this machinery we give new streamlined proofs of some classical finiteness theorems in algebraic statistics as well as a proof of the independent set conjecture of hosten and the second author

recursive parametrization and invariant phases of unitary matrices we present further properties of a previously proposed recursive scheme for parametrization of n by n unitary matrices we show that the factors in the recursive formula may be introduced in any desired order the method is used to study the invariant phases of unitary matrices the case of four by four unitary matrices is investigated in detail we also address the question of how to construct symmetric unitary matrices i e unitary matrices u that satisfy the condition uij uji using the recursive approach

some applications of quasisymmetry summary we give three applications of quasisymmetry the first two applications involve derived conditional distributions in a study involving matched individuals where the parameters associated with the matching are eliminated by appropriate conditioning extensions to matched triplets or to larger blocks lead to generalizations of quasisymmetry the third application involves reversibility of markov chains where the connexion between reversibility and quasisymmetry is exploited to construct tests for reversibility and tests for homogeneity of several reversible chains

sequential monte carlo p values summary the assessment of statistical significance by monte carlo simulation may be costly in computer time this paper looks at a number of ways of calculating exact monte carlo p values by sequential sampling such p values are shown to have properties similar to those obtained by sampling with a fixed sample size both standard and generalized monte carlo procedures are discussed and in particular a sequential method is proposed for dealing with situations in which values can only be conveniently generated using a markov chain conditioned to pass through the observed data

an exact mcnemar test for paired binary markov chains a straightforward extension of the mcnemar test for paired binary data yields an exact test for the equality of the limiting marginal distributions for bivariate binary markov chains the exact distribution of the test statistics under the null hypothesis of equal marginals depends on the classical cell occupancy statistics for the bose einstein model exact p values are computed for the one sided test and the mean and variance of the test statistic are found the power of the markov mcnemar test is found to be close to the power of the classical mcnemar test for independent paired observations when the independence assumption holds the method is applied to the comparison of ribosomal dna sequences