the geometry of the universal teichmuller space and the euler weil petersson equation on the identity component of the universal teichmuller space endowed with the takhtajan teo topology the geodesics of the weil petersson metric are shown to exist for all time this component is naturally a subgroup of the quasisymmetric homeomorphisms of the circle viewed this way the regularity of its elements is shown to be h3 2 epsilon for all epsilon 0 the evolutionary pde associated to the spatial representation of the geodesics of the weil petersson metric is derived using multiplication and composition below the critical sobolev index 3 2 geodesic completeness is used to introduce special classes of solutions of this pde analogous to peakons our setting is used to prove that there exists a unique geodesic between each two shapes in the plane in the context of the application of the weil petersson metric in imaging c 2015 elsevier inc all rights reserved

finite dimensional irreducible representations of the quantum superalgebra u q gl n 1 it is shown that every finite dimensional irreducible module over the general linear lie superalgebragl n 1 can be deformed to an irreducible module ofu q gl n 1 aq analogue of the universal enveloping algebra ofgl n 1 the results are extended also to all kac modules which in the atypical cases remain indecomposible within each module expressions for the transformations of the gel fand zetlin basis under the action of the algebra generators are written down an analogoue of the poincare birkhoff witt theorem is formulated

q oscillator realizations of the metaplectic representation of quantum osp 3 2 q oscillator representations of the quantum superalgebra ospq 3 2 are discussed it is pointed out that the metaplectic representation of ospq 3 2 can be equivalently realized either in terms of one q boson and one q fermion or with one q boson and two q fermions

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

composite kernel machine regression based on likelihood ratio test for joint testing of genetic and gene environment interaction effect most common human diseases are a result from the combined effect of genes the environmental factors and their interactions such that including gene environment ge interactions can improve power in gene mapping studies the standard strategy is to test the snps one by one using a regression model that includes both the snp effect and the ge interaction however the snp by snp approach has serious limitations such as the inability to model epistatic snp effects biased estimation and reduced power thus in this article we develop a kernel machine regression framework to model the overall genetic effect of a snp set considering the possible ge interaction specifically we use a composite kernel to specify the overall genetic effect via a nonparametric function andwe model additional covariates parametrically within the regression framework the composite kernel is constructed as a weighted average of two kernels one corresponding to the genetic main effect and one corresponding to the ge interaction effect we propose a likelihood ratio test lrt and a restricted likelihood ratio test rlrt for statistical significance we derive a monte carlo approach for the finite sample distributions of lrt and rlrt statistics extensive simulations and real data analysis show that our proposed method has correct type i error and can have higher power than score based approaches under many situations

q deformed oscillator associated with the calogero model and its q coherent state we present the q deformed para bose oscillators associated with the two body calogero model the q deformed coherent state is also constructed and its resolution of unity is demonstrated

supersymmetry parastatistics and operator realizations of a lie algebra the algebraic structure of parastatistics has been generalized and it is found to be consistent with supersymmetric quantum mechanics with supercharges constructed out of the generalized para bose and para fermi operators it is further shown that the operator algebra of generalized parastatistics offers a realization of the graded orthosymplectic group similar to that of orthogonal and symplectic groups using conventional parastatistics

virasoro model space the representations of a compact lie groupg can be studied via the construction of an associated model space this space has the property that when geometrically quantized its hilbert space contains every irreducible representation ofg just once we construct an analogous space for the group diffs1 it is naturally a complex manifold with a holomorphic free action of diffs1 preserving a family of pseudo kahler structures all of the good coadjoint orbits are obtained from our space by hamiltonian constraint reduction we briefly discuss the connection to the work of alekseev and shatashvili

geometry and representations of the quantum supergroup ospq 1 2n the quantum supergroup ospq 1 2n is studied systematically a haar functional is constructed and an algebraic version of the peter weyl theory is extended to this quantum supergroup quantum homogeneous superspaces and quantum homogeneous supervector bundles are defined following the strategy of connes theory parabolic induction is developed by employing the quantum homogeneous supervector bundles quantum frobenius reciprocity and a generalized borel weil theorem are established for the induced representations

bayesian semiparametric regression models for evaluating pathway effects on continuous and binary clinical outcomes many statistical methods for microarray data analysis consider one gene at a time and they may miss subtle changes at the single gene level this limitation may be overcome by considering a set of genes simultaneously where the gene sets are derived from prior biological knowledge limited work has been carried out in the regression setting to study the effects of clinical covariates and expression levels of genes in a pathway either on a continuous or on a binary clinical outcome hence we propose a bayesian approach for identifying pathways related to both types of outcomes we compare our bayesian approaches with a likelihood based approach that was developed by relating a least squares kernel machine for nonparametric pathway effect with a restricted maximum likelihood for variance components unlike the likelihood based approach the bayesian approach allows us to directly estimate all parameters and pathway effects it can incorporate prior knowledge into bayesian hierarchical model formulation and makes inference by using the posterior samples without asymptotic theory we consider several kernels gaussian polynomial and neural network kernels to characterize gene expression effects in a pathway on clinical outcomes our simulation results suggest that the bayesian approach has more accurate coverage probability than the likelihood based approach and this is especially so when the sample size is small compared with the number of genes being studied in a pathway we demonstrate the usefulness of our approaches through its applications to a type ii diabetes mellitus data set our approaches can also be applied to other settings where a large number of strongly correlated predictors are present

field models from two cocycles on infinite dimensional lie groups and symplectic structures 2d gravity and chern simons theory in this paper the connection between field models and infinite dimensional lie groups is widely analysed on the bases of a new group quantization approach we also relate the poincare cartan form of variational calculus to the symplectic current structure of the covariant phase space formulation of higher derivative field theory the virasoro and kac moody groups are considered in the first case the action functional of the 2d induced gravity in the light cone formulation is derived the hidden sl 2 r simply appears as generated by the kernel of the lie algebra two cocycle and plays the role of a gauge type symmetry nevertheless it is shown that a proper space like formulation is out of reach of the virasoro group the corresponding symplectic structure of the nonlocal action functional is determined showing that it is related to the symplectic structure associated with the sl 2 r kac moody group this unravels the proper geometrical meaning of the hidden symmetry and differs from the analysis in related works based on the coadjoint orbit approach the relation between the kac moody groups and the chern simons gauge theory on a disc in the presence of a source is considered using the new approach

linear score tests for variance components in linear mixed models and applications to genetic association studies summary following the rapid development of genome scale genotyping technologies genetic association mapping has become a popular tool to detect genomic regions responsible for certain disease phenotypes especially in early phase pharmacogenomic studies with limited sample size in response to such applications a good association test needs to be 1 applicable to a wide range of possible genetic models including but not limited to the presence of gene by environment or gene by gene interactions and non linearity of a group of marker effects 2 accurate in small samples fast to compute on the genomic scale and amenable to large scale multiple testing corrections and 3 reasonably powerful to locate causal genomic regions the kernel machine method represented in linear mixed models provides a viable solution by transforming the problem into testing the nullity of variance components in this study we consider score based tests by choosing a statistic linear in the score function when the model under the null hypothesis has only one error variance parameter our test is exact in finite samples when the null model has more than one variance parameter we develop a new moment based approximation that performs well in simulations through simulations and analysis of real data we demonstrate that the new test possesses most of the aforementioned characteristics especially when compared to existing quadratic score tests or restricted likelihood ratio tests

quantum dynamics on super groups constraints and the brst supergroup the smallest supergroup b k containing among its generators those of a constraint group k the brst charge and the corresponding ghost and antighost operators is constructed the supergroup b k can be enlarged to include the ordinary dynamical variables of the unconstrained physical system in a group approach to quantization b k generalizes the ordinary u 1 phase invariance of wave functions in particular this mechanism reproduces the essential features of the brst quantization when k is diff s1 the critical values of string theory are those for which the space of polarized functions on the enlarged brst supergroup is not irreducible

aspects of infinite dimensional l super galilean conformal algebra in this work we construct an infinite dimensional l super galilean conformal algebra which is a generalization of the l 1 algebra found in the literature we give a classification of central extensions the vector field representation the coadjoint representation and the operator product expansion of the infinite dimensional l super galilean conformal algebra keeping possible applications in physics and mathematics in mind

dynamics on sl 2 r tilde x u 1 a complete analysis of the quantum dynamical system defined by the quantum group sl 2 r tilde x u 1 is made this quantum group which is primarily related to the relativistic harmonic oscillator is also shown to provide the quantum dynamics of a free particle moving in 1 1 dimensional anti de sitter spacetime the latter interpretation illustrates the capabilities of the present group approach to quantisation in dealing with dynamical systems in more general spacetimes

symmetry and quantization higher order polarization and anomalies the concept of full polarization subalgebra in a group approach to quantization on a lie group g as a generalization of the analogous concept in geometric or standard quantization is discussed the lack of full polarization subalgebras is considered as an anomaly of the corresponding system and related to its more conventional definition a generalization of the subalgebra of full polarization is then provided made out of higher order differential operators in the enveloping algebra of g higher order polarizations can also be used to quantize nonanomalous theories in different representations numerous examples are analyzed including the finite dimensional dynamics associated with the schrodinger group which presents an anomaly and an infinite dimensional anomalous system associated with the virasoro group in the last example the operators in the higher order polarization are in one to one correspondence with the null vectors in the verma module approach

comparison of maximum statistics for hypothesis testing when a nuisance parameter is present only under the alternative in many practical problems a hypothesis testing involves a nuisance parameter which appears only under the alternative hypothesis davies 1977 biometrika 64 247 254 proposed the maximum of the score statistics over the whole range of the nuisance parameter as a test statistic for this type of hypothesis testing freidlin podgor and gastwirth 1999 biometrics 55 883 886 studied two other simpler maximum test statistics the maximum of the score statistics at two extreme points of the nuisance parameter and the maximum of the score statistics at three points of the nuisance parameter including the two extreme points in this article we compare the powers of these three maximum type statistics in the context of three genetic problems

quantization as a consequence of the symmetry group an approach to geometric quantization a method is proposed to obtain the dynamics of a system which only makes use of the group law it incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics the classical or quantum character of the theory is related to the choice of the group avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems i e the classical limit the method is applied to the free particle dynamics and the harmonic oscillator

testing and estimation in marker set association study using semiparametric quantile regression kernel machine we consider quantile regression for partially linear models where an outcome of interest is related to covariates and a marker set e g gene or pathway the covariate effects are modeled parametrically and the marker set effect of multiple loci is modeled using kernel machine we propose an efficient algorithm to solve the corresponding optimization problem for estimating the effects of covariates and also introduce a powerful test for detecting the overall effect of the marker set our test is motivated by traditional score test and borrows the idea of permutation test our estimation and testing procedures are evaluated numerically and applied to assess genetic association of change in fasting homocysteine level using the vitamin intervention for stroke prevention trial data

sdiff s2 and the orbit method the group of area preserving diffeomorphisms of the two sphere sdiff s2 is one of the simplest examples of an infinite dimensional lie group it plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related newly defined symmetry groups we investigate its representation theory using the method of coadjoint orbits we describe the casimir functions and the cartan algebra then we evaluate the trace of a simple sdiff s2 operator using the atiyah bott fixed point formula the trace is divergent but we show that it has well defined truncations related to the structure of sdiff s2 finally we relate our results back to the recent appearances of sdiff s2 in black hole physics the group of area preserving diffeomorphisms of the two sphere sdiff s2 is one of the simplest examples of an infinite dimensional lie group it plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related newly defined symmetry groups we investigate its representation theory using the method of coadjoint orbits we describe the casimir functions and the cartan algebra then we evaluate the trace of a simple sdiff s2 operator using the atiyah bott fixed point formula the trace is divergent but we show that it has well defined truncations related to the structure of sdiff s2 finally we relate our results back to the recent appearances of sdiff s2 in black hole physics

q oscillator realizations of the quantum superalgebras slq m n andospq m 2n realizations of the quantum superalgebras corresponding to thea m n b m n c n 1 andd m n series are given in terms of the creation and annihilation operators ofq deformed bose and fermi oscillators

on a general framework for q particles paraparticles and q paraparticles through deformations the authors propose a unification of deformations associated with the descriptions of q para and q para bosons and fermions for complex values of q

on quantization of simple harmonic oscillators a general framework is presented of quantizing the system of a simple harmonic oscillator various methods of quantization so far proposed are accommodated here in a unified form some remarks are also made on quantization for the case of more than one oscillator

representations of osp 2 1 and the metaplectic representation shift operator techniques are used to treat the irreducible representations of the superalgebra osp 2 1 apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact su 2 algebra the case when the star conditions on the even part are those satisfied by the noncompact su 1 1 algebra is also treated in this case no gradestar representations arise and the star representations are found to consist of the direct sum of two discrete series representations of su 1 1 one of these representations can be realized in terms of functions of a single complex variable and turns out to be a simple example of a metaplectic representation

characters of the bms group in three dimensions using the frobenius formula we evaluate characters associated with certain induced representations of the centrally extended bms3 group this computation involves a functional integral over a coadjoint orbit of the virasoro group a delta function localizes the integral to a single point allowing us to obtain an exact result the latter is independent of the specific form of the functional measure and holds for all values of the bms3 central charges and all values of the chosen mass and spin it can also be recovered as a flat limit of virasoro characters

parabosons virasoro type algebras and their deformations two parameter p q deformation of a parabosonic algebra underlying the two particle calogero model is considered the fock space representation bargmann fock representation coherent states spectrum generating algebra and a hidden supersymmetry are discussed the paraboson algebra induces a centreless virasoro type algebra with a doubling of the space of generators a two parameter deformation of this virasoro type algebra is studied

highest weight irreducible representations of the lie superalgebra gl 1 two classes of irreducible highest weight modules of the general linear lie superalgebra gl 1 are constructed within each module a basis is introduced and the transformation relations of the basis under the action of the algebra generators are written down

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

quantum deformation of the ladder representations of u 1 1 for q 1 uq u 1 1 covariant creation and annihilation operators for q on the unit circle are constructed and the corresponding ladder and singleton representations are considered

short distance expansion from the dual representation of infinite dimensional lie algebras we develop a method for computing the short distance expansion of fields or operators that live in the coadjoint representation of an infinite dimensional lie algebra by using only properties of the adjoint representation and its dual we explicitly implement this method by computing the short distance expansion for the duals of the virasoro algebra affine lie algebras and the geometrically realized n extended supersymmetric virasoro algebra this method can also be used to compute short distance expansions between fields that transform in the adjoint and those that transform in the coadjoint representations

coadjoint orbits of the virasoro group the coadjoint orbits of the virasoro group which have been investigated by lazutkin and pankratova and by segal should according to the kirillov kostant theory be related to the unitary representations of the virasoro group in this paper the classification of orbits is reconsidered with an explicit description of the possible centralizers of coadjoint orbits the possible orbits are diff s1 itself diff s1 s1 and diff s1 sl n 2 r withsl n 2 r a certain discrete series of embeddings ofsl 2 r in diff s1 and diffs1 t wheret may be any of certain rather special one parameter subgroups of diffs1 an attempt is made to clarify the relation between orbits and representations it appears that quantization of diffs1 s1 is related to unitary representations with nondegenerate kac determinant unitary verma modules while quantization of diffs1 sl n 2 r is seemingly related to unitary representations with null vectors in leveln a better understanding of how to quantize the relevant orbits might lead to a better geometrical understanding of virasoro representation theory in the process of investigating virasoro coadjoint orbits we observe the existence of left invariant symplectic structures on the virasoro group manifold as is described in an appendix these give rise to lie bialgebra structures with the virasoro algebra as the underlying lie algebra

the relativistic bargmann transform an integral transformation generalizing the bargmann transform is established among two sets of wavefunctions associated with the configuration and bargmann fock segal like representations of a relativistic harmonic oscillator the time independent configuration space wavefunctions are also studied and the lack of unitarity occurring when factorizing the time dependence of the wavefunction is solved by modifying the scalar product

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

construction of local one parameter subgroups generated by the virasoro operators via feynman path integrals abstract it is well known that the virasoro algebra acts on the representations of an infinite dimensional heisenberg algebra called the fock models we calculate feynman path integrals on an coadjoint orbit of the infinite dimensional heisenberg group to give explicit formulae of actions of local one parameters subgroups generated by the virasoro operators

increasing power for voxel wise genome wide association studies the random field theory least square kernel machines and fast permutation procedures imaging traits are thought to have more direct links to genetic variation than diagnostic measures based on cognitive or clinical assessments and provide a powerful substrate to examine the influence of genetics on human brains although imaging genetics has attracted growing attention and interest most brain wide genome wide association studies focus on voxel wise single locus approaches without taking advantage of the spatial information in images or combining the effect of multiple genetic variants in this paper we present a fast implementation of voxel and cluster wise inferences based on the random field theory to fully use the spatial information in images the approach is combined with a multi locus model based on least square kernel machines to associate the joint effect of several single nucleotide polymorphisms snp with imaging traits a fast permutation procedure is also proposed which significantly reduces the number of permutations needed relative to the standard empirical method and provides accurate small p value estimates based on parametric tail approximation we explored the relation between 448 294 single nucleotide polymorphisms and 18 043 genes in 31 662 voxels of the entire brain across 740 elderly subjects from the alzheimer s disease neuroimaging initiative adni structural mri scans were analyzed using tensor based morphometry tbm to compute 3d maps of regional brain volume differences compared to an average template image based on healthy elderly subjects we find method to be more sensitive compared with voxel wise single locus approaches a number of genes were identified as having significant associations with volumetric changes the most associated gene was grin2b which encodes the n methyl d aspartate nmda glutamate receptor nr2b subunit and affects both the parietal and temporal lobes in human brains its role in alzheimer s disease has been widely acknowledged and studied suggesting the validity of the approach the various advantages over existing approaches indicate a great potential offered by this novel framework to detect genetic influences on human brains

higher order polarizations on the virasoro group and anomalies in a previous paper the authors showed that the space of first order polarized functions on the virasoro group is not in general irreducible the full reduction was explicitly achieved by taking the orbit of the enveloping algebra through the vacuum this additional step provided the proper quantization in the strong coupling domain 0 c 1 in this paper we introduce the concept of higher order polarization as a generalization of that of polarization we prove that the imposing of the additional higher order polarization conditions is equivalent to the taking of the above mentioned orbit this demonstrates that the generalized higher order polarization conditions suffice to obtain the irreducible hilbert spaces we also discuss the need for higher order polarizations in terms of anomalies

testing against a high dimensional alternative in the generalized linear model asymptotic type i error control summary testing a low dimensional null hypothesis against a high dimensional alternative in a generalized linear model may lead to a test statistic that is a quadratic form in the residuals under the null model using asymptotic arguments we show that the distribution of such a test statistic can be approximated by a ratio of quadratic forms in normal variables for which algorithms are readily available for generalized linear models the asymptotic distribution shows good control of type i error for moderate to small samples even when the number of covariates in the model far exceeds the sample size

generalized reduced rank latent factor regression for high dimensional tensor fields and neuroimaging genetic applications we propose a generalized reduced rank latent factor regression model grrlf for the analysis of tensor field responses and high dimensional covariates the model is motivated by the need from imaging genetic studies to identify genetic variants that are associated with brain imaging phenotypes often in the form of high dimensional tensor fields grrlf identifies from the structure in the data the effective dimensionality of the data and then jointly performs dimension reduction of the covariates dynamic identification of latent factors and nonparametric estimation of both covariate and latent response fields after accounting for the latent and covariate effects grllf performs a nonparametric test on the remaining factor of interest grrlf provides a better factorization of the signals compared with common solutions and is less susceptible to overfitting because it exploits the effective dimensionality the generality and the flexibility of grrlf also allow various statistical models to be handled in a unified framework and solutions can be efficiently computed within the field of neuroimaging it improves the sensitivity for weak signals and is a promising alternative to existing approaches the operation of the framework is demonstrated with both synthetic datasets and a real world neuroimaging example in which the effects of a set of genes on the structure of the brain at the voxel level were measured and the results compared favorably with those from existing approaches

a description of the superalgebra osp 2 n 1 2 m via green generators an alternative description of the lie superalgebra osp 2n 1 2m in terms of generators and relations is given the generators called green generators are the root vectors of osp 2n 1 2m corresponding to the orthogonal roots

irreducible representations of the lie superalgebras b 0 n with finite degenerated vacuum ii the problem of getting irreducible representations of lie superalgebras b 0 n n 1 2 is studied starting with a recently constructed family of linear representations in terms of differential operators on the space c n of cn valued c functions equivalent formulation via creation annihilation operators of a para bose system with n degrees of freedom is used and the domain d of any is shown to be a subset of c n containing a nonzero vacuum subspace by assuming its dimension finite the necessary conditions for existence of are derived the method is applied to the superalgebra b 0 1 and a one parameter family of nonequivalent irreducible representations in terms of unbounded linear operators on l2 r c2 is obtained each representation has a nondegenerated vacuum and for all z b 0 1 satisfying z z the operators z are essentially self adjoint

the quantum superalgebra uq osp 1 2n deformed para bose operators and root of unity representations we recall the relation between the lie superalgebra osp 1 2n and para bose operators the quantum superalgebra uq osp 1 2n defined as usual in terms of its chevalley generators is shown to be isomorphic to an associative algebra generated by so called pre oscillator operators satisfying a number of relations from these relations and the analogue with the non deformed case one can interpret these pre oscillator operators as deformed para bose operators some consequences for uq osp 1 2n cartan weyl basis poincare birkhoff win basis and its hopf subalgebra uq gl n are pointed out finally using a realization in terms of q commuting q bosons we construct an irreducible finite dimensional unitary fock representation of uq osp 1 2n and its decomposition in terms of uq gl n representations when q is a root of unity

characterizing the function space for bayesian kernel models kernel methods have been very popular in the machine learning literature in the last ten years mainly in the context of tikhonov regularization algorithms in this paper we study a coherent bayesian kernel model based on an integral operator defined as the convolution of a kernel with a signed measure priors on the random signed measures correspond to prior distributions on the functions mapped by the integral operator we study several classes of signed measures and their image mapped by the integral operator in particular we identify a general class of measures whose image is dense in the reproducing kernel hilbert space rkhs induced by the kernel a consequence of this result is a function theoretic foundation for using non parametric prior specifications in bayesian modeling such as gaussian process and dirichlet process prior distributions we discuss the construction of priors on spaces of signed measures using gaussian and levy processes with the dirichlet processes being a special case the latter computational issues involved with sampling from the posterior distribution are outlined for a univariate regression and a high dimensional classification problem

kernel machine approach to testing the significance of multiple genetic markers for risk prediction summary there is growing evidence that genomic and proteomic research holds great potentials for changing irrevocably the practice of medicine the ability to identify important genomic and biological markers for risk assessment can have a great impact in public health from disease prevention to detection to treatment selection however the potentially large number of markers and the complexity in the relationship between the markers and the outcome of interest impose a grand challenge in developing accurate risk prediction models the standard approach to identifying important markers often assesses the marginal effects of individual markers on a phenotype of interest when multiple markers relate to the phenotype simultaneously via a complex structure such a type of marginal analysis may not be effective to overcome such difficulties we employ a kernel machine cox regression f ramework and propose an efficient score test to assess the overall effect of a set of markers such as genes within a pathway or a network on survival outcomes the proposed test has the advantage of capturing the potentially non linear effects without explicitly specifying a particular non linear functional form to approximate the null distribution of the score statistic we propose a simple resampling procedure that can be easily implemented in practice numerical studies suggest that the test performs well with respect to both empirical size and power even when the number of variables in a gene set is not small compared to the sample size

quantization of uq osp 1 2n with deformed para bose operators the observation that n pairs of para bose pb operators generate the universal enveloping algebra of the orthosymplectic lie superalgebra osp 1 2n is used in order to define deformed pb operators it is shown that these operators are an alternative to the chevalley generators on this background uq osp 1 2n its cartan weyl generators and their supercommutation relations are written down entirely in terms of deformed pb operators an analogue of the poincare birkhoff witt theorem is formulated

r matrix formulation of deformed boson algebra rewriting the pusz woronowicz q boson relations in terms of the standard r matrix for suq n the author gives the definition of a general deformed boson algebra a r depending upon a matrix r he investigates the conditions under which this algebra is associative for n 2 a set of matrices satisfying these conditions is classified and the corresponding twisted statistics is given

algebraic quantization on a group and nonabelian constraints a generalization of a previous group manifold quantization formalism is proposed in the new version the differential structure is circumvented so that discrete transformations in the group are allowed and a nonabelian group replaces the ordinary central u 1 subgroup of the heisenberg weyl like quantum group as an example of the former we obtain the wave functions associated with the system of two identical particles and the latter modification is used to account for the virasoro constraints in string theory

variants of bosonization in parabosonic algebra the hopf and super hopf structures in parabosonic algebra parabosonic algebra in finite or infinite degrees of freedom is considered as a 22 graded associative algebra and is shown to be a 22 graded or super hopf algebra the super hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the osp 1 2n lie superalgebraic structure the notion of super hopf algebra is equivalently described as a hopf algebra in the braided monoidal category cz 2 m the bosonization technique for switching a hopf algebra in the braided monoidal category h m where his a quasitriangular hopf algebra into an ordinary hopf algebra is reviewed in this paper we prove that for the parabosonic algebra p b beyond the application of the bosonization technique to the original super hopf algebra a bosonization like construction is also achieved using two operators related to the parabosonic total number operator both techniques switch the same super hopf algebra p b to an ordinary hopf algebra thus producing two different variants of p b with an ordinary hopf structure

smoothing spline models with correlated random errors abstract spline smoothing techniques are commonly used to estimate the mean function in a nonparametric regression model their performances depend greatly on the choice of smoothing parameters many methods of selecting smoothing parameters such as generalized maximum likelihood gml generalized cross validation gcv and unbiased risk ubr have been developed under the assumption of independent observations they tend to underestimate smoothing parameters when data are correlated in this article i assume that observations are correlated and that the correlation matrix depends on a parsimonious set of parameters i extend the gml gcv and ubr methods to estimate the smoothing parameters and the correlation parameters simultaneously i also relate a smoothing spline model to three mixed effects models these relationships show that the smoothing spline estimates evaluated at design points are best linear unbiased prediction blup estimates and that the gml estimates of the smoothing parameters an

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

a superalgebra uq osp 3 2 generated by deformed paraoperators and its morphism onto a wq 1 1 clifford weyl algebra a deformed universal enveloping superalgebra uq of the lie superalgebra osp 3 2 which is a hopf algebra is defined as freely generated by deformed para bose operators b and deformed para fermi operators f the q analog of the cartan weyl basis and all deformed supercommutation relations between the generators are written down a morphism of uq onto a deformed clifford weyl superalgebra wq 1 1 a is constructed where wq 1 1 a is generated by q bosons b and q fermions f under the condition that the q bosons anticommute with the q fermions the corresponding oscillator representation is studied

new solutions of the yang baxter equation based on root of 1 representations of the para bose superalgebra new solutions of the quantum yang baxter equation depending in general on three arbitrary parameters are written down they are based on the root of unity representations of the quantum orthosymplectic superalgebra which were found recently representations of the braid group are defined within any nth tensorial power of root of 1 modules

algebraic structure of parabose fock space i the green s ansatz revisited a description of the fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed a new statement of the well known green s ansatz is introduced to help understand the significance of the bilinear commutation relations it is also applied to show how all parabose oscillators of even order respectively odd order can be obtained as irreducible constituents of the fock space associated with a green s ansatz of two three terms analogs of the form in which the parabose green s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order methods used in the paper allow various new results for q deformed parabose oscillators to be derived

invariant measures on polarized submanifolds in group quantization we provide an explicit construction of quasi invariant measures on polarized coadjoint orbits of a lie group g the use of specific trivial central extensions of g by the multiplicative group r allows us to restore the strict invariance of the measures and accordingly the unitarity of the quantization of coadjoint orbits as an example the representations of sl 2 r are recovered

unitarizable representations of the deformed para bose superalgebra uq osp 1 2 at roots of 1 the unitarizable irreps of the deformed para bose superalgebra pbq which is isomorphic to uq osp 1 2 are classified at q being root of 1 new finite dimensional irreps of uq osp 1 2 are found explicit expressions for the matrix elements are written down

a nonparametric smoothing method for assessing gee models with longitudinal binary data studies involving longitudinal binary responses are widely applied in the health and biomedical sciences research and frequently analyzed by generalized estimating equations gee method this article proposes an alternative goodness of fit test based on the nonparametric smoothing approach for assessing the adequacy of gee fitted models which can be regarded as an extension of the goodness of fit test of le cessie and van houwelingen biometrics 1991 47 1267 1282 the expectation and approximate variance of the proposed test statistic are derived the asymptotic distribution of the proposed test statistic in terms of a scaled chi squared distribution and the power performance of the proposed test are discussed by simulation studies the testing procedure is demonstrated by two real data

performance of variable and function selection methods for estimating the nonlinear health effects of correlated chemical mixtures a simulation study statistical methods for identifying harmful chemicals in a correlated mixture often assume linearity in exposure response relationships nonmonotonic relationships are increasingly recognized eg for endocrine disrupting chemicals however the impact of nonmonotonicity on exposure selection has not been evaluated in a simulation study we assessed the performance of bayesian kernel machine regression bkmr bayesian additive regression trees bart bayesian structured additive regression with spike slab priors bstarss generalized additive models with double penalty gamdp and thin plate shrinkage smoothers gamts multivariate adaptive regression splines mars and lasso penalized regression we simulated realistic exposure data based on pregnancy exposure to 17 phthalates and phenols in the us national health and nutrition examination survey using a multivariate copula we simulated data sets of size n 250 and compared methods across 32 scenarios varying by model size and sparsity signal to noise ratio correlation structure and exposure response relationship shapes we compared methods in terms of their sensitivity specificity and estimation accuracy in most scenarios bkmr bstarss gamdp and gamts achieved moderate to high sensitivity 0 52 0 98 and specificity 0 21 0 99 bart and mars achieved high specificity 0 90 but low sensitivity in low signal to noise ratio scenarios 0 20 0 51 lasso was highly sensitive 0 71 0 99 except for quadratic relationships 0 27 penalized regression methods that assume linearity such as lasso may not be suitable for studies of environmental chemicals hypothesized to have nonmonotonic relationships with outcomes instead bkmr bstarss gamdp and gamts are attractive methods for flexibly estimating the shapes of exposure response relationships and selecting among correlated exposures

random effects selection in linear mixed models nous presentons le probleme pratique important de la selection des effets aleatoires dans un modele lineaire mixte un modele bayesien hierarchique est utilise pour identifier tout effet aleatoire devariance nulle l approche proposee reparametrise le modele mixte de telle sorte que les fonctions des parametres de covariance de la distribution des effets aleatoires soient incorporees comme des coefficients de regression pour des variables latentes normales nous permettons aux effets aleatoires de sortir du modele en choisissant des melanges d a priori avec des masses ponctuelles en zero pour les variances des effets aleatoires du fait de la reparametrisation le modele prend une structure lineaire conditionnelle qui facilite l utilisation de lois a priori conjuguees et normales nous demontrons que les calculs a posteriori peuvent etre effectues via un algorithme simple et efficace de monte carlo par chaine de markov les methodes sont illustrees par des donnees simulees et par des donnees reelles d exposition prenatale au polychlorobiphenile et des donnees sur le developpement psychomoteur des enfants

semiparametric stochastic mixed models for longitudinal data abstract we consider inference for a semiparametric stochastic mixed model for longitudinal data this model uses parametric fixed effects to represent the covariate effects and an arbitrary smooth function to model the time effect and accounts for the within subject correlation using random effects and a stationary or nonstationary stochastic process we derive maximum penalized likelihood estimators of the regression coefficients and the nonparametric function the resulting estimator of the nonparametric function is a smoothing spline we propose and compare frequentist inference and bayesian inference on these model components we use restricted maximum likelihood to estimate the smoothing parameter and the variance components simultaneously we show that estimation of all model components of interest can proceed by fitting a modified linear mixed model we illustrate the proposed method by analyzing a hormone dataset and evaluate its performance through simulations

quantum deformation of bose parastatistics a q deformation of the transformation of the chevalley basis to an odd basis of generators of the universal enveloping of the lie superalgebra g n osp 1 2n is presented it is shown that one thus obtains a reasonable quantum deformation of the algebra of n para bose oscillators

canonical coherent states for the relativistic harmonic oscillator in this paper there are constructed manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given sl 2 r representation once a change of variables z c zd unit disk is performed also introduced are higher order relativistic creation and annihilation operators a a with canonical commutation relation a a 1 rather than the covariant one z z energy and naturally associated with the sl 2 r group the canonical relativistic coherent states are then defined as eigenstates of a finally a canonical minimal representation is constructed in configuration space by means of eigenstates of a canonical position operator

non abelian harmonic oscillators and chiral theories we show that a large class of physical theories which has been under intensive investigation recently share the same geometric features in their hamiltonian formulation these dynamical systems range from harmonic oscillations to wzw like models and to the kdv dynamics on diff0s1 to the same class belong also the hamiltonian systems on groups of maps the common feature of these models are the chiral equations of motion for so called chiral decomposition of the phase space

q analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

a description of the quantum superalgebrauq sl n 1 m via creation and annihilation generators a description of the quantum superalgebra uq sl n 1 m and in particular of the special linear superalgebra sl n 1 m via creation and annihilation generators cags is given it provides an alternative to the canonical description of uq sl n 1 m in terms of chevalley generators a conjecture that the fock representations of the cags provide microscopic realizations of exclusion statistics is formulated

formal group laws for affine kac moody groups and group quantization we describe a method for obtaining formal group laws from the structure constants of affine kac moody groups and then apply a group manifold quantization procedure which permits construction of physical representations by using only canonical structures on the group as an intermediate step we get an explicit expression for two cocycles on loop groups the programme is applied to the affinesu 2 group

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

equivalence of kernel machine regression and kernel distance covariance for multidimensional phenotype association studies summary associating genetic markers with a multidimensional phenotype is an important yet challenging problem in this work we establish the equivalence between two popular methods kernel machine regression kmr and kernel distance covariance kdc kmr is a semiparametric regression framework that models covariate effects parametrically and genetic markers non parametrically while kdc represents a class of methods that include distance covariance dc and hilbert schmidt independence criterion hsic which are nonparametric tests of independence we show that the equivalence between the score test of kmr and the kdc statistic under certain conditions can lead to a novel generalization of the kdc test that incorporates covariates our contributions are 3 fold 1 establishing the equivalence between kmr and kdc 2 showing that the principles of kmr can be applied to the interpretation of kdc 3 the development of a broader class of kdc statistics where the class members are statistics corresponding to different kernel combinations finally we perform simulation studies and an analysis of real data from the alzheimer s disease neuroimaging initiative adni study the adni study suggest that snps of flj16124 exhibit pairwise interaction effects that are strongly correlated to the changes of brain region volumes

semiparametric regression of multidimensional genetic pathway data least squares kernel machines and linear mixed models we consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least squares kernel machines lskms this unified framework allows a flexible function for the joint effect of multiple genes within a pathway by specifying a kernel function and allows for the possibility that each gene expression effect might be nonlinear and the genes within the same pathway are likely to interact with each other in a complicated way this semiparametric model also makes it possible to test for the overall genetic pathway effect we show that the lskm semiparametric regression can be formulated using a linear mixed model estimation and inference hence can proceed within the linear mixed model framework using standard mixed model software both the regression coefficients of the covariate effects and the lskm estimator of the genetic pathway effect can be obtained using the best linear unbiased predictor in the corresponding linear mixed model formulation the smoothing parameter and the kernel parameter can be estimated as variance components using restricted maximum likelihood a score test is developed to test for the genetic pathway effect model variable selection within the lskm framework is discussed the methods are illustrated using a prostate cancer data set and evaluated using simulations

polynomial deformations of osp 1 2 and generalized parabosons we consider the algebra r generated by three elements a b h subject to three relations h a a h b b and a b f h when f h h this coincides with the lie superalgebra osp 1 2 when f is a polynomial we speak of polynomial deformations of osp 1 2 irreducible representations of r are described and in the case deg f 2 we obtain a complete classification showing some similarities but also some interesting differences with the usual osp 1 2 representations the relation with deformed oscillator algebras is discussed leading to the interpretation of r as a generalized paraboson algebra

symplectic manifolds coherent states and semiclassical approximation the symplectic structure and hamiltonian dynamics for a class of grassmannian manifolds are described here using the two dimensional sphere s2 and disc d2 as illustrative cases we write their path integral representations using coherent state techniques these path integrals can be evaluated exactly by semiclassical methods thus providing examples of the localization formula along the way we also give a local coordinate description for a class of grassmannians

quantization symmetry and natural polarization we discuss the notion of polarization as defined in a geometric quantization scheme recently introduced in terms of the role played by the evolution operator of the quantum system the analysis uses an integral transform representation of the group wsp 2 r this clarifies the group theoretic origin of the natural polarizations and the meaning of the polarization changing transformations

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

uq covariant oscillators and vertex operators uq sl 2 covariant bose and fermi oscillators are studied systematically and applied to constructing vertex operators and uq invariant blocks for qp 1 p 2 3 a p2 dimensional unitary fock space representation of the bose oscillator algebra a q is constructed this representation does not allow however for the construction of all uq invariant blocks associated with an indecomposable representation of uq an extension of a is needed in this case which involves operators creating states of q spin projection p 2 from the fock vacuum the extended algebra only has infinite dimensional indefinite metric space representations an algebraic notion of trace is introduced provided q2 1 in the bose case this allows the introduction of a family of in general mixed uq invariant temperature states such that the vacuum expectation value appears in the zero temperature limit

fock representations of the superalgebra sl n 1 m its quantum analogue uq sl n 1 m and related quantum statistics fock space representations of the lie superalgebra sl n 1 m and of its quantum analogue uq sl n 1 m are written down the results are based on a description of these superalgebras via creation and annihilation operators the properties of the underlying statistics are briefly discussed

bayesian varying coefficient kernel machine regression to assess neurodevelopmental trajectories associated with exposure to complex mixtures exposure to environmental mixtures can exert wide ranging effects on child neurodevelopment however there is a lack of statistical methods that can accommodate the complex exposure response relationship between mixtures and neurodevelopment while simultaneously estimating neurodevelopmental trajectories we introduce bayesian varying coefficient kernel machine regression bvckmr a hierarchical model that estimates how mixture exposures at a given time point are associated with health outcome trajectories the bvckmr flexibly captures the exposure response relationship incorporates prior knowledge and accounts for potentially nonlinear and nonadditive effects of individual exposures this model assesses the directionality and relative importance of a mixture component on health outcome trajectories and predicts health effects for unobserved exposure profiles using contour plots and cross sectional plots bvckmr also provides information about interactions between complex mixture components the bvckmr is applied to a subset of data from progress a prospective birth cohort study in mexico city on exposure to metal mixtures and temporal changes in neurodevelopment the mixture include metals such as manganese arsenic cobalt chromium cesium copper lead cadmium and antimony results from a subset of programming research in obesity growth environment and social stressors data provide evidence of significant positive associations between second trimester exposure to copper and bayley scales of infant and toddler development cognition score at 24 months and cognitive trajectories across 6 24 months we also detect an interaction effect between second trimester copper and lead exposures for cognition at 24 months in summary bvckmr provides a framework for estimating neurodevelopmental trajectories associated with exposure to complex mixtures

quantum computation as gravity we formulate nielsen s geometric approach to circuit complexity in the context of two dimensional conformal field theories where series of conformal transformations are interpreted as unitary circuits built from energy momentum tensor gates we show that the complexity functional in this setup can be written as the polyakov action of two dimensional gravity or equivalently as the geometric action on the coadjoint orbits of the virasoro group this way we argue that gravity sets the rules for optimal quantum computation in conformal field theories

hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the jordan schwinger map the aim of this paper is to show that there is a hopf structure of the parabosonic and parafermionic algebras and this hopf structure can generate the well known hopf algebraic structure of the lie algebras through a realization of lie algebras using the parabosonic and parafermionic extension of the jordan schwinger map the differences between the hopf algebraic and the graded hopf superalgebraic structure on the parabosonic algebra are discussed