adaptive bayesian wavelet shrinkage abstract when fitting wavelet based models shrinkage of the empirical wavelet coefficients is an effective tool for denoising the data this article outlines a bayesian approach to shrinkage obtained by placing priors on the wavelet coefficients the prior for each coefficient consists of a mixture of two normal distributions with different standard deviations the simple and intuitive form of prior allows us to propose automatic choices of prior parameters these parameters are chosen adaptively according to the resolution level of the coefficients typically shrinking high resolution frequency coefficients more heavily assuming a good estimate of the background noise level we obtain closed form expressions for the posterior means and variances of the unknown wavelet coefficients the latter may be used to assess uncertainty in the reconstruction several examples are used to illustrate the method and comparisons are made with other shrinkage methods

quasistationary trajectories of the mean field xy hamiltonian model a topological perspective we employ a topological approach to investigate the nature of quasistationary states of the mean field xy hamiltonian model we focus on the quasistationary states reached when the system is initially prepared in a fully magnetized configuration by means of numerical simulations and analytical considerations we show that along the quasistationary trajectories the system evolves in a manifold of critical points of the potential energy function although these critical points are maxima the large number of directions with marginal stability may be responsible for the slow relaxation dynamics and the trapping of the system in such trajectories

configurations in abelian categories ii ringel hall algebras abstract this is the second in a series on configurations in an abelian category a given a finite poset i an i configuration is a finite collection of objects j and morphisms j k or j k j k in a satisfying some axioms where j k i configurations describe how an object x in a decomposes into subobjects the first paper defined configurations and studied moduli spaces of i configurations in a using the theory of artin stacks it showed well behaved moduli stacks obj a m i a of objects and configurations in a exist when a is the abelian category coh p of coherent sheaves on a projective scheme p or mod k q of representations of a quiver q write cf obj a for the vector space of q valued constructible functions on the stack obj a motivated by the idea of ringel hall algebras we define an associative multiplication on cf obj a using pushforwards and pullbacks along 1 morphisms between configuration moduli stacks so that cf obj a is a q algebra we also study representations of cf obj a the lie subalgebra cf ind obj a of functions supported on indecomposables and other algebraic structures on cf obj a then we generalize all these ideas to stack functions sf obj a a universal generalization of constructible functions containing more information when ext i x y 0 for all x y a and i 1 or when a coh p for p a calabi yau 3 fold we construct lie algebra morphisms from stack algebras to explicit algebras which will be important in the sequels on invariants counting semistable objects in a

stationary point approach to the phase transition of the classical xy chain with power law interactions the stationary points of the potential energy function v of the classical xy chain with power law pair interactions i e interactions decaying like r with the distance are analyzed for a class of spinwave type stationary points the asymptotic behavior of the hessian determinant of v is computed analytically in the limit of large system size the computation is based on the toeplitz property of the hessian and makes use of a szego type theorem the results serve to illustrate a recently discovered relation between phase transitions and the properties of stationary points of classical many body potentials in agreement with this relation the exact phase transition potential energy of the model can be read off from the behavior of the hessian determinant for exponents between zero and one for between one and two the phase transition is not manifest in the behavior of the determinant and it might be necessary to consider larger classes of stationary points

virtual refinements of the vafa witten formula we conjecture a formula for the generating function of virtual chi y genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2 form specializing the conjecture to minimal surfaces of general type and to virtual euler characteristics we recover part of a formula of c vafa and e witten these virtual chi y genera can be written in terms of descendent donaldson invariants using t mochizuki s formula the latter can be expressed in terms of seiberg witten invariants and certain explicit integrals over hilbert schemes of points these integrals are governed by seven universal functions which are determined by their values on mathbb p 2 and mathbb p 1 times mathbb p 1 using localization we calculate these functions up to some order which allows us to check our conjecture in many cases in an appendix by h nakajima and the first named author the virtual euler characteristic specialization of our conjecture is extended to include mu classes thereby interpolating between vafa witten s formula and witten s conjecture for donaldson invariants

modeling microarray data using a threshold mixture model an important goal of microarray studies is the detection of genes that show significant changes in expression when two classes of biological samples are being compared we present an anova style mixed model with parameters for array normalization overall level of gene expression and change of expression between the classes for the latter we assume a mixing distribution with a probability mass concentrated at zero representing genes with no changes and a normal distribution representing the level of change for the other genes we estimate the parameters by optimizing the marginal likelihood to make this practical laplace approximations and a backfitting algorithm are used the performance of the model is studied by simulation and by application to publicly available data sets

a general maximum likelihood analysis of variance components in generalized linear models this paper describes an em algorithm for nonparametric maximum likelihood ml estimation in generalized linear models with variance component structure the algorithm provides an alternative analysis to approximate mql and pql analyses mcgilchrist and aisbett 1991 biometrical journal 33 131 141 breslow and clayton 1993 journal of the american statistical association 88 9 25 mcgilchrist 1994 journal of the royal statistical society series b 56 61 69 goldstein 1995 multilevel statistical models and to gee analyses liang and zeger 1986 biometrika 73 13 22 the algorithm first given by hinde and wood 1987 in longitudinal data analysis 110 126 is a generalization of that for random effect models for overdispersion in generalized linear models described in aitkin 1996 statistics and computing 6 251 262 the algorithm is initially derived as a form of gaussian quadrature assuming a normal mixing distribution but with only slight variation it can be used for a completely unknown mixing distribution giving a straightforward method for the fully nonparametric ml estimation of this distribution this is of value because the ml estimates of the glm parameters can be sensitive to the specification of a parametric form for the mixing distribution the nonparametric analysis can be extended straightforwardly to general random parameter models with full npml estimation of the joint distribution of the random parameters this can produce substantial computational saving compared with full numerical integration over a specified parametric distribution for the random parameters a simple method is described for obtaining correct standard errors for parameter estimates when using the em algorithm several examples are discussed involving simple variance component and longitudinal models and small area estimation

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

energy landscape analysis of the two dimensional nearest neighbor φ4 model the stationary points of the potential energy function of the a 4 model on a two dimensional square lattice with nearest neighbor interactions are studied by means of two numerical methods a numerical homotopy continuation method and a globally convergent newton raphson method we analyze the properties of the stationary points in particular with respect to a number of quantities that have been conjectured to display signatures of the thermodynamic phase transition of the model although no such signatures are found for the nearest neighbor a 4 model our study illustrates the strengths and weaknesses of the numerical methods employed

partial equivalence of statistical ensembles and kinetic energy abstract the phenomenon of partial equivalence of statistical ensembles is illustrated by discussing two examples the mean field xy and the mean field spherical model the configurational parts of these systems exhibit partial equivalence of the microcanonical and the canonical ensemble furthermore the configurational microcanonical entropy is a smooth function whereas a nonanalytic point of the configurational free energy indicates the presence of a phase transition in the canonical ensemble in the presence of a standard kinetic energy contribution partial equivalence is removed and a nonanalyticity arises also microcanonically hence in contrast to the common belief kinetic energy even though a quadratic form in the momenta has a nontrivial effect on the thermodynamic behaviour as a by product we present the microcanonical solution of the mean field spherical model with kinetic energy for finite and infinite system sizes

exploring the energy landscape of xy models we investigate the energy landscape of two and three dimensional xy models with nearest neighbor interactions by analytically constructing several classes of stationary points of the hamiltonian these classes are analyzed in particular with respect to possible signatures of the thermodynamic phase transitions of the models we find that even after explicitly breaking the global o 2 symmetry of the xy spins an exponentially large class of stationary points are singular and occur in continuous one parameter families this property may complicate the use of theoretical tools developed for the investigation of phase transitions based on stationary points of the energy landscape and we discuss strategies to avoid these difficulties

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

standard errors for em estimates in generalized linear models with random effects a procedure is derived for computing standard errors of em estimates in generalized linear models with random effects quadrature formulas are used to approximate the integrals in the em algorithm where two different approaches are pursued i e gauss hermite quadrature in the case of gaussian random effects and nonparametric maximum likelihood estimation for an unspecified random effect distribution an approximation of the expected fisher information matrix is derived from an expansion of the em estimating equations this allows for inferential arguments based on em estimates as demonstrated by an example and simulations

equivariant k theory and refined vafa witten invariants in maulik and thomas in preparation the vafa witten theory of complex projective surfaces is lifted to oriented mathbb c equivariant cohomology theories here we study the k theoretic refinement it gives rational functions in t 1 2 invariant under t 1 2 leftrightarrow t 1 2 which specialise to numerical vafa witten invariants at t 1 on the instanton branch the invariants give the virtual chi t genus refinement of gottsche kool extended to allow for strictly semistable sheaves applying modularity to their calculations gives predictions for the contribution of the monopole branch we calculate some cases and find perfect agreement we also do calculations on k3 surfaces finding jacobi forms refining the usual modular forms proving a conjecture of gottsche kool we determine the k theoretic virtual classes of degeneracy loci using eagon northcott complexes and show they calculate refined vafa witten invariants using this laarakker monopole contributions to refined vafa witten invariants arxiv 1810 00385 proves universality results for the invariants

configurations in abelian categories iii stability conditions and identities this is the third in a series on configurations in an abelian category given a finite poset i an i configuration s p is a finite collection of objects s j and morphisms j k or in satisfying some axioms where j k are subsets of i configurations describe how an object x in decomposes into subobjects the first paper defined configurations and studied moduli spaces of configurations in using the theory of artin stacks it showed well behaved moduli stacks of objects and configurations in exist when is the abelian category coh p of coherent sheaves on a projective scheme p or of representations of a quiver q the second studied algebras of constructible functions and stack functions on this paper introduces weak stability conditions t t on we show the moduli spaces of t semistable indecomposable t semistable and t stable objects in class a are constructible sets in and some associated configuration moduli spaces constructible in so their characteristic functions and are constructible we prove many identities relating these constructible functions and their stack function analogues under pushforwards we introduce interesting algebras of constructible and stack functions and study their structure in the fourth paper we show are independent of t t and construct invariants of

degeneracy loci classes in k theory determinantal and pfaffian formula abstract we prove a determinantal formula that describes the k theoretic degeneracy loci classes for grassmann bundles we further prove pfaffian formulas for symplectic and odd orthogonal grassmann bundles the former generalizes damon kempf laksov s determinantal formula and the latter generalize pragacz kazarian s formulas for the chow ring

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

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

donaldson thomas type invariants via microlocal geometry we prove that donaldson thomas type invariants are equal to weighted euler characteristics of their moduli spaces in particular such invariants depend only on the scheme structure of the moduli space not the symmetric obstruction theory used to define them we also introduce new invariants generalizing donaldson thomas type invariants to moduli problems with open moduli space these are useful for computing donaldson thomas type invariants over stratifications

nested hilbert schemes on surfaces virtual fundamental class abstract we construct virtual fundamental classes on nested hilbert schemes of points and curves in complex nonsingular projective surfaces these classes recover the virtual classes of seiberg witten theory as well as the reduced stable theory and play a crucial role in the reduced donaldson thomas theory of local surface threefolds that we study in 15 we show that certain integrals against the virtual fundamental classes of punctual nested hilbert schemes are expressed as integrals over the products of the hilbert scheme of points we are able to find explicit formulas for some of these integrals by relating them to carlsson okounkov s vertex operator formulas

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

wavelet thresholding via a bayesian approach we discuss a bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression a prior distribution is imposed on the wavelet coefficients of the unknown response function designed to capture the sparseness of wavelet expansion that is common to most applications for the prior specified the posterior median yields a thresholding procedure our prior model for the underlying function can be adjusted to give functions falling in any specific besov space we establish a relationship between the hyperparameters of the prior model and the parameters of those besov spaces within which realizations from the prior will fall such a relationship gives insight into the meaning of the besov space parameters moreover the relationship established makes it possible in principle to incorporate prior knowledge about the function s regularity properties into the prior model for its wavelet coefficients however prior knowledge about a function s regularity properties might be difficult to elicit with this in mind we propose a standard choice of prior hyperparameters that works well in our examples several simulated examples are used to illustrate our method and comparisons are made with other thresholding methods we also present an application to a data set that was collected in an anaesthesiological study