properties of the error linear complexity spectrum this paper studies the error linear complexity spectrum of binary sequences with period 2n a precise categorization of those sequences having two distinct critical points in their spectra as well as an enumeration of these sequences is given an upper bound on the maximum number of distinct critical points that the spectrum of a sequence can have is proved and a construction which yields a lower bound on this number is given in the process simpler proofs of some known results on the linear complexity and k error linear complexity of sequences with period 2n are provided

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

bayesian sparse multiple regression for simultaneous rank reduction and variable selection we develop a bayesian methodology aimed at simultaneously estimating low rank and row sparse matrices in a high dimensional multiple response linear regression model we consider a carefully devised shrinkage prior on the matrix of regression coefficients which obviates the need to specify a prior on the rank and shrinks the regression matrix towards low rank and row sparse structures we provide theoretical support to the proposed methodology by proving minimax optimality of the posterior mean under the prediction risk in ultra high dimensional settings where the number of predictors can grow sub exponentially relative to the sample size a one step post processing scheme induced by group lasso penalties on the rows of the estimated coefficient matrix is proposed for variable selection with default choices of tuning parameters we additionally provide an estimate of the rank using a novel optimization function achieving dimension reduction in the covariate space we exhibit the performance of the proposed methodology in an extensive simulation study and a real data example

bayesian factorizations of big sparse tensors it has become routine to collect data that are structured as multiway arrays tensors there is an enormous literature on low rank and sparse matrix factorizations but limited consideration of extensions to the tensor case in statistics the most common low rank tensor factorization relies on parallel factor analysis parafac which expresses a rank k tensor as a sum of rank one tensors in contingency table applications in which the sample size is massively less than the number of cells in the table the low rank assumption is not sufficient and parafac has poor performance we induce an additional layer of dimension reduction by allowing the effective rank to vary across dimensions of the table taking a bayesian approach we place priors on terms in the factorization and develop an efficient gibbs sampler for posterior computation theory is provided showing posterior concentration rates in high dimensional settings and the methods are shown to have excellent performance in simulations and several

an algorithm for the k error linear complexity of binary sequences with period 2 sup n certain applications require pseudo random sequences which are unpredictable in the sense that recovering more of the sequence from a short segment must be computationally infeasible it is shown that linear complexity is useful in determining such sequences a generalized linear complexity that has application to the security of stream ciphers is proposed and an efficient algorithm is given for the case in which the sequence is binary with period 2 sup n this algorithm generalizes an algorithm presented by r a games and a h chan 1983

computing the error linear complexity spectrum of a binary sequence of period 2 sup n binary sequences with high linear complexity are of interest in cryptography the linear complexity should remain high even when a small number of changes are made to the sequence the error linear complexity spectrum of a sequence reveals how the linear complexity of the sequence varies as an increasing number of the bits of the sequence are changed we present an algorithm which computes the error linear complexity for binary sequences of period spl lscr 2 sup n using o spl lscr log spl lscr sup 2 bit operations the algorithm generalizes both the games chan 1983 and stamp martin 1993 algorithms which compute the linear complexity and the k error linear complexity of a binary sequence of period spl lscr 2 sup n respectively we also discuss an application of an extension of our algorithm to decoding a class of linear subcodes of reed muller codes

investigation of painleve property under time singularities transformations in this paper the relation between the painleve property for ordinary differential equations and some coordinate transformations in the complex plane is investigated two sets of transformations are introduced for which the transformations of singularities structure in the complex plane are explicitly computed the first set acts only on the dependent variables while the second one acts also on the independent variables it allows for defining an extended painleve test which is coordinates invariant our results are applied on various problems arising in the singularity analysis such as the problem of negative resonances and the weak painleve conjecture the method for finding new first integrals for dynamical systems is also shown

correlation immunity of nonlinear combining functions for cryptographic applications corresp pseudonoise generators for cryptographic applications consisting of several linear feedback shift registers with a nonlinear combining function have been proposed as running key generators in stream ciphers these running key generators eau sometimes be broken by ciphertext only correlation attacks on individual subsequences a new class of combining functions is presented which provides better security against such attacks the security is quantified by the smallest number m 1 of subsequences that must be simultaneously considered in a correlation attack a necessary condition for such m th order correlation immunity is proved a recursive construction is given that permits the construction of an m th order immune combining function for n subsequences for any m and n with 1 leq m finally the trade off between the length of the linear equivalent of the nonlinear generator and the order m of its immunity against correlation attacks is considered

new covering radius of reed muller codes for t resilient functions in this paper we introduce a new covering radius of rm r n from cryptography viewpoint it is defined as the maximum distance between t resilient functions and the rth order reed muller code rm r n we next derive its lower and upper bounds we further present a table of numerical data of our bounds

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

on some inequalities for the incomplete gamma function let p 1 be a positive real number we determine all real numbers p and p such that the inequalities formula math formula math are valid for all x 0 and we determine all real numbers a and b such that log 1 e ax x e t t log 1 e bx hold for all 0

painleve property and geometry abstract the painleve property for discrete hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times the augmented manifold is constructed by expanding the hamilton jacobi equation a complete classification of the types of poles allowed in complex time is given for hamiltonians which separate into the direct product of hyperelliptic curves for such systems bounds on the degrees of the polynomial separating variable change and the other integrals in involution can be found from the pole series and the hamilton jacobi equation it is shown how branching can arise naturally in a painleve system

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

the bayesian lasso the lasso estimate for linear regression parameters can be interpreted as a bayesian posterior mode estimate when the regression parameters have independent laplace i e double exponential priors gibbs sampling from this posterior is possible using an expanded hierarchy with conjugate normal priors for the regression parameters and independent exponential priors on their variances a connection with the inverse gaussian distribution provides tractable full conditional distributions the bayesian lasso provides interval estimates bayesian credible intervals that can guide variable selection moreover the structure of the hierarchical model provides both bayesian and likelihood methods for selecting the lasso parameter slight modifications lead to bayesian versions of other lasso related estimation methods including bridge regression and a robust variant

the algebraic complete integrability of geodesic flow on so n we study for which left invariant diagonal metrics onso n the euler arnold equations dot x x lambda x x x ij in so n lambda ij x ij lambda ij lambda ji can be linearized on an abelian variety i e are solvable by quadratures we show that merely by requiring that the solutions of the differential equations be single valued functions of complex timet suffices to prove that under a non degeneracy assumption on the metric the only such metrics are those which satisfy manakov s conditions ij bi bj ai aj 1 the case of degenerate metrics is also analyzed forn 4 this provides a new and simpler proof of a result of adler and van moerbeke 3

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

mirror transformations of hamiltonian systems we demonstrate through various examples of hamiltonian systems that symplectic structures have been encoded into the painleve test each principal balance in the painleve test induces a mirror transformation that regularizes movable singularities moreover for finite dimensional hamiltonian systems the mirror transformations are canonical

recursive lower bounds on the nonlinearity profile of boolean functions and their applications the nonlinearity profile of a boolean function i e the sequence of its minimum hamming distances nlr f to all functions of degrees at most r for r ges 1 is a cryptographic criterion whose role against attacks on stream and block ciphers has been illustrated by many papers it plays also a role in coding theory since it is related to the covering radii of reed muller codes we introduce a method for lower bounding its values and we deduce bounds on the second order nonlinearity for several classes of cryptographic boolean functions including the welch and the multiplicative inverse functions used in the s boxes of the advanced encryption standard aes in the case of this last infinite class of functions we are able to bound the whole profile and we do it in an efficient way when the number of variables is not too small this allows showing the good behavior of this function with respect to this criterion as well

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

fast sampling with gaussian scale mixture priors in high dimensional regression we propose an efficient way to sample from a class of structured multivariate gaussian distributions the proposed algorithm only requires matrix multiplications and linear system solutions its computational complexity grows linearly with the dimension unlike existing algorithms that rely on cholesky factorizations with cubic complexity the algorithm is broadly applicable in settings where gaussian scale mixture priors are used on high dimensional parameters its effectiveness is illustrated through a high dimensional regression problem with a horseshoe prior on the regression coefficients other potential applications are outlined

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

nonlinear evolution equations of hamiltonian form and the painleve analysis a number of necessary conditions for a class of nonlinear partial differential equations to pass the painleve test with the kruskal ansatz is given a theory for the admissible resonance patterns of equations in hamiltonian form is developed and discussed for some important special cases based on these results on algorithm can be described which constructs all nonlinear evolution equations of normal type and a certain hamiltonian form

best affine and quadratic approximations of particular classes of boolean functions in this paper we consider the problem of computing best low order approximations of boolean functions we focus on the best quadratic approximations of a subclass of cubic functions with arbitrary number of variables and we provide formulas for their efficient calculation our methodology is developed upon properties of the best affine approximations of quadratic functions for which formulas for their direct computation not by means of the walsh hadamard transform are given we determine the cubic functions in the above subclass that achieve the maximum second order nonlinearity thus yielding a lower bound for the covering radius of the second order reed muller code ssr rm 2 n in ssr rm 3 n simple extensions of these results to some special cases of higher degree functions are seen to hold furthermore a preliminary analysis of well known constructions for bent functions in terms of their second order nonlinearity is performed that indicates potential weaknesses if construction parameters are not properly chosen

bayes and empirical bayes multiplicity adjustment in the variable selection problem this paper studies the multiplicity correction effect of standard bayesian variable selection priors in linear regression our first goal is to clarify when and how multiplicity correction happens automatically in bayesian analysis and to distinguish this correction from the bayesian ockham s razor effect our second goal is to contrast empirical bayes and fully bayesian approaches to variable selection through examples theoretical results and simulations considerable differences between the two approaches are found in particular we prove a theorem that characterizes a surprising aymptotic discrepancy between fully bayes and empirical bayes this discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical bayes estimate indeed even at the extreme when the empirical bayes estimate converges asymptotically to the true variable inclusion probability the potential for a serious difference remains

elastic net regression modeling with the orthant normal prior the elastic net procedure is a form of regularized optimization for linear regression that provides a bridge between ridge regression and the lasso the estimate that it produces can be viewed as a bayesian posterior mode under a prior distribution implied by the form of the elastic net penalty this article broadens the scope of the bayesian connection by providing a complete characterization of a class of prior distributions that generate the elastic net estimate as the posterior mode the resulting model based framework allows for methodology that moves beyond exclusive use of the posterior mode by considering inference based on the full posterior distribution two characterizations of the class of prior distributions are introduced a properly normalized direct characterization which is shown to be conjugate for linear regression models and an alternate representation as a scale mixture of normal distributions prior distributions are proposed for the regularization parameters resulting in an infi

bernstein von mises theorems for gaussian regression with increasing number of regressors this paper brings a contribution to the bayesian theory of nonparametric and semiparametric estimation we are interested in the asymptotic normality of the posterior distribution in gaussian linear regression models when the number of regressors increases with the sample size two kinds of bernstein von mises theorems are obtained in this framework nonparametric theorems for the parameter itself and semiparametric theorems for functionals of the parameter we apply them to the gaussian sequence model and to the regression of functions in sobolev and c classes in which we get the minimax convergence rates adaptivity is reached for the bayesian estimators of functionals in our applications

high dimensional covariance matrix estimation in approximate factor models the variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems classical methods of estimating the covariance matrices are based on the strict factor models assuming independent idiosyncratic components this assumption however is restrictive in practical applications by assuming sparse error covariance matrix we allow the presence of the cross sectional correlation even after taking out common factors and it enables us to combine the merits of both methods we estimate the sparse covariance using the adaptive thresholding technique as in cai and liu j amer statist assoc 106 2011 672 684 taking into account the fact that direct observations of the idiosyncratic components are unavailable the impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

integrating the nonintegrable analytic structure of the lorenz system revisited abstract a study of the complex time analytic structure of the lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one time dependent integral of motion exists furthermore the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument the analytic techniques used in these studies a systematic resummation of a logarithmic psi series appears to be quite general and can provide explicit representations of a solution even in the chaotic regimes in the neighborhood of a given movable singularity furthermore we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

a relationship between linear complexity and k error linear complexity linear complexity is an important cryptographic criterion of stream ciphers the k error linear complexity of a periodic sequence of period n is defined as the smallest linear complexity that can be obtained by changing k or fewer bits of the sequence per period this article shows a relationship between the linear complexity and the minimum value k for which the k error linear complexity is strictly less than the linear complexity

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

high dimensional sparse factor modeling applications in gene expression genomics we describe studies in molecular profiling and biological pathway analysis that use sparse latent factor and regression models for microarray gene expression data we discuss breast cancer applications and key aspects of the modeling and computational methodology our case studies aim to investigate and characterize heterogeneity of structure related to specific oncogenic pathways as well as links between aggregate patterns in gene expression profiles and clinical biomarkers based on the metaphor of statistically derived factors as representing biological subpathway structure we explore the decomposition of fitted sparse factor models into pathway subcomponents and investigate how these components overlay multiple aspects of known biological activity our methodology is based on sparsity modeling of multivariate regression anova and latent factor models as well as a class of models that combines all components hierarchical sparsity priors address questions of dimension reduction and multiple co

expandable factor analysis bayesian sparse factor models have proven useful for characterizing dependence in multivariate data but scaling computation to large numbers of samples and dimensions is problematic we propose expandable factor analysis for scalable inference in factor models when the number of factors is unknown the method relies on a continuous shrinkage prior for efficient maximum a posteriori estimation of a low rank and sparse loadings matrix the structure of the prior leads to an estimation algorithm that accommodates uncertainty in the number of factors we propose an information criterion to select the hyperparameters of the prior expandable factor analysis has better false discovery rates and true positive rates than its competitors across diverse simulation settings we apply the proposed approach to a gene expression study of ageing in mice demonstrating superior results relative to four competing methods

a fast algorithm for determining the complexity of a binary sequence with period 2 n corresp the complexity of a periodic sequence s is defined to be the least number of stages in a linear feedback shift register that generates s

factor modeling for high dimensional time series inference for the number of factors this paper deals with the factor modeling for high dimensional time series based on a dimension reduction viewpoint under stationary settings the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix and is therefore applicable when the dimension of time series is on the order of a few thousands asymptotic properties of the proposed method are investigated under two settings i the sample size goes to infinity while the dimension of time series is fixed and ii both the sample size and the dimension of time series go to infinity together in particular our estimators for zero eigenvalues enjoy faster convergence or slower divergence rates hence making the estimation for the number of factors easier in particular when the sample size and the dimension of time series go to infinity together the estimators for the eigenvalues are no longer consistent however our estimator for the number of the factors which is based on the ratios of the estimated eigenvalues still works fine furthermore this estimation shows the so called blessing of dimensionality property in the sense that the performance of the estimation may improve when the dimension of time series increases a two step procedure is investigated when the factors are of different degrees of strength numerical illustration with both simulated and real data is also reported

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

integrability partial integrability and nonintegrability for systems of ordinary differential equations the integrability of systems of ordinary differential equations with polynomial vector fields is investigated by using the singularity analysis methods three types of results are obtained first a general relationship between the degrees of first integrals and the so called kowalevskaya exponents is derived second it is shown that all solutions of algebraically integrable systems can be expanded in puiseux series third a new method to study partially integrable systems is studied these different aspects allow us to study algorithmically the integrability partial integrability and nonintegrability of differential systems

lie symmetries of the lorenz model abstract we study the generalized symmetries of the lorenz model to find the parameter values at which one or more time dependent integrals of motion exist in these cases the integrals are found trivially from the symmetries themselves a complete study of the one completely algebraically integrable case shows a the dynamics can be integrated exactly by reducing it first to a lower dimensional system b the symmetry vector field is hamiltonian these facts hold for other dissipative completely integrable dynamical systems as well the analytic study of a natural two form reveals that it is an entire function of time the foliation of phase space induced by the two form for the partially integrable cases has a simple description in terms of the coefficients occurring in the laurent series expansions of the dependent variables

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

the error linear complexity spectrum as a cryptographic criterion of boolean functions the error linear complexity spectrum constitutes a well known cryptographic criterion for sequences indicating how the linear complexity of the sequence decreases as the number of bits allowed to be modified per period increases in this paper via defining an association between 2 n periodic binary sequences and boolean functions on n variables it is shown that the error linear complexity spectrum also provides useful cryptographic information for the corresponding boolean function f namely it yields an upper bound on the minimum hamming distance between f and the set of functions depending on fewer number of variables therefore the prominent lauder paterson algorithm for computing the error linear complexity spectrum of a sequence may also be used for efficiently determining approximations of a boolean function that depend on fewer number of variables moreover it is also shown that through this approach low degree approximations of a boolean function can be also obtained in an efficient way

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

the horseshoe estimator for sparse signals this paper proposes a new approach to sparsity called the horseshoe estimator the horseshoe is a close cousin of other widely used bayes rules arising from for example double exponential and cauchy priors in that it is a member of the same family of multivariate scale mixtures of normals but the horseshoe enjoys a number of advantages over existing approaches including its robustness its adaptivity to dierent sparsity patterns and its analytical tractability we prove two theorems that formally characterize both the horseshoe s adeptness at large outlying signals and its super ecient rate of convergence to the correct estimate of the sampling density in sparse situations finally using a combination of real and simulated data we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full bayesian model averaging approach using a discrete mixture prior to model signals and noise

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

fast bayesian factor analysis via automatic rotations to sparsity abstractrotational post hoc transformations have traditionally played a key role in enhancing the interpretability of factor analysis regularization methods also serve to achieve this goal by prioritizing sparse loading matrices in this work we bridge these two paradigms with a unifying bayesian framework our approach deploys intermediate factor rotations throughout the learning process greatly enhancing the effectiveness of sparsity inducing priors these automatic rotations to sparsity are embedded within a pxl em algorithm a bayesian variant of parameter expanded em for posterior mode detection by iterating between soft thresholding of small factor loadings and transformations of the factor basis we obtain a dramatic accelerations b robustness against poor initializations and c better oriented sparse solutions to avoid the prespecification of the factor cardinality we extend the loading matrix to have infinitely many columns with the indian buffet process ibp prior the factor dimen

on the structure of picard fuchs type equations for liouville arnold integrable hamiltonian system on cotangent phase spaces there are studied in detail the structure properties of integral submanifold imbedding mapping for a class of algebraically liouville integrable hamiltonian systems on cotangent phase spaces and related with it so called picard fuchs type equations it is shown that these equations can be in general regularly constructed making use of a given a priori system of involutive invariants and proved that their solutions in the hamolton jacobi separable variable case give rise exactly to the integral submanifold imbedding mapping being as known a main ingredient for liouville arnold integrability by quadratures of the hamiltonian system under regard

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series