fine structure of spectral properties for random correlation matrices an application to financial markets we study some properties of eigenvalue spectra of financial correlation matrices in particular we investigate the nature of the large eigenvalue bulks which are observed empirically and which have often been regarded as a consequence of the supposedly large amount of noise contained in financial data we challenge this common knowledge by acting on the empirical correlation matrices of two data sets with a filtering procedure which highlights some of the cluster structure they contain and we analyze the consequences of such filtering on eigenvalue spectra we show that empirically observed eigenvalue bulks emerge as superpositions of smaller structures which in turn emerge as a consequence of cross correlations between stocks we interpret and corroborate these findings in terms of factor models and we compare empirical spectra to those predicted by random matrix theory for such models

eigenvalue significance testing for genetic association genotype eigenvectors are widely used as covariates for control of spurious stratification in genetic association significance testing for the accompanying eigenvalues has typically been based on a standard tracy widom limiting distribution for the largest eigenvalue derived under white noise assumptions it is known that even modest local correlation among markers inflates the largest eigenvalues even in the absence of true stratification in addition a few sample eigenvalues may be extreme creating further complications in accurate testing we explore several methods to identify appropriate null eigenvalue thresholds while remaining sensitive to eigenvalues corresponding to population stratification we introduce a novel block permutation approach designed to produce an appropriate null eigenvalue distribution by eliminating long range genomic correlation while preserving local correlation we also propose a fast approach based on eigenvalue distribution modeling using a simple fit criterion and the general marcenko pastur equation under a simple discrete eigenvalue model block permutation and the model based approach work well for pure simulations and for data resampled from the 1000 genomes project in contrast we find that the standard approach of computing an effective number of markers does not perform well the performance of the methods is also demonstrated for a motivating example from the international cystic fibrosis consortium

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

hardy spaces and divergence operators on strongly lipschitz domains of rn abstract let be a strongly lipschitz domain of r n consider an elliptic second order divergence operator l including a boundary condition on and define a hardy space by imposing the non tangential maximal function of the extension of a function f via the poisson semigroup for l to be in l1 under suitable assumptions on l we identify this maximal hardy space with h 1 r n if r n with h r 1 under the dirichlet boundary condition and with h z 1 under the neumann boundary condition

irregularity of hypergeometric systems via slopes along coordinate subspaces we study the irregularity sheaves attached to the a hypergeometric d module ma introduced by gel fand et al where a z d n is pointed of full rank and c more precisely we investigate the slopes of this module along coordinate subspaces in the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector l on torus equivariant generators to this end we introduce the a l umbrella a simplicial complex determined by a and l and identify its facets with the components of the associated graded ring we then establish a correspondence between the full a l umbrella and the components of the l characteristic variety of ma we compute in combinatorial terms the multiplicities of these components in the l characteristic cycle of the associated euler koszul complex identifying them with certain intersection multiplicities we deduce from this that slopes of ma are combinatorial independent of and in one to one correspondence with jumps of the a l umbrella this confirms a conjecture of sturmfels and gives a converse of a theorem of hotta ma is regular if and only if a defines a projective variety 1991 mathematics subject classification 13n10 14m25 16w70

tent space boundedness via extrapolation we study the action of operators on tent spaces such as maximal operators calderon zygmund operators riesz potentials we also consider singular non integral operators we obtain boundedness as an application of extrapolation methods in the banach range in the non banach range boundedness results for calderon zygmund operators follow by using an appropriate atomic theory we end with some consequences on amalgam spaces

singularities and holonomicity of binomial d modules abstract we study binomial d modules which generalize a hypergeometric systems we determine explicitly their singular loci and provide three characterizations of their holonomicity the first of these is an equivalence of holonomicity and l holonomicity for these systems the second refines the first by giving more detailed information about the l characteristic variety of a non holonomic binomial d module the final characterization states that a binomial d module is holonomic if and only if its corresponding singular locus is proper

scaling in the eigenvalue fluctuations of correlation matrices the spectra of empirical correlation matrices constructed from multivariate data are widely used in many areas of sciences engineering and social sciences as a tool to understand the information contained in typically large datasets in the last two decades random matrix theory based tools such as the nearest neighbour eigenvalue spacing and eigenvector distributions have been employed to extract the significant modes of variability present in such empirical correlations in this work we present an alternative analysis in terms of the recently introduced spacing ratios which does not require the cumbersome unfolding process it is shown that the higher order spacing ratio distributions for the wishart ensemble of random matrices characterized by the dyson index beta is related to the first order spacing ratio distribution with a modified value of co dimension beta this scaling is demonstrated for wishart ensemble and also for the spectra of empirical correlation matrices drawn from the observed stock market and atmospheric pressure data using a combination of analytical and numerics such scalings in spacing distributions are also discussed

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

weak type p p estimates for riesz transforms we consider the problem of l p boundedness of higher order riesz transforms associated to elliptic operators l of order 2m on as an application of the recently solved kato conjecture we show for all this generalizes the result of auscher and tchamitchian restricted to the case d 2m

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

new function spaces of bmo type the john nirenberg inequality interpolation and applications in this paper we introduce and develop some new function spaces of bmo bounded mean oscillation type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type the new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity and they generalize the classical bmo space we show that the john nirenberg inequality holds on these spaces and they interpolate with lp spaces by the complex interpolation method we then give applications on lp boundedness of singular integrals whose kernels do not satisfy the hormander condition 2005 wiley periodicals inc

hypergeometric functions for projective toric curves abstract we produce a decomposition of the parameter space of the a hypergeometric system associated to a projective monomial curve as a union of an arrangement of lines and its complement in such a way that the analytic behavior of the solutions of the system is explicitly controlled within each term of the union

cluster structure in the correlation coefficient matrix can be characterized by abnormal eigenvalues abstract in a large number of previous studies the researchers found that some of the eigenvalues of the financial correlation matrix were greater than the predicted values of the random matrix theory r m t here we call these eigenvalues as abnormal eigenvalues in order to reveal the hidden meaning of these abnormal eigenvalues we study the toy model with cluster structure and find that these eigenvalues are related to the cluster structure of the correlation coefficient matrix in this paper model based experiments show that in most cases the number of abnormal eigenvalues of the correlation matrix is equal to the number of clusters in addition empirical studies show that the sum of the abnormal eigenvalues is related to the clarity of the cluster structure and is negatively correlated with the correlation dimension

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality