two sample testing in high dimensions summary we propose new methodology for two sample testing in high dimensional models the methodology provides a high dimensional analogue to the classical likelihood ratio test and is applicable to essentially any model class where sparse estimation is feasible sparse structure is used in the construction of the test statistic in the general case testing then involves non nested model comparison and we provide asymptotic results for the high dimensional setting we put forward computationally efficient procedures based on data splitting including a variant of the permutation test that exploits sparse structure we illustrate the general approach in two sample comparisons of high dimensional regression models differential regression and graphical models differential network showing results on simulated data as well as data from two recent cancer studies

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

a multivariate two sample mean test for small sample size and missing data we develop a new statistic for testing the equality of two multivariate mean vectors a scaled chi squared distribution is proposed as an approximating null distribution because the test statistic is based on componentwise statistics it has the advantage over hotelling s t2 test of being applicable to the case where the dimension of an observation exceeds the number of observations an appealing feature of the new test is its ability to handle missing data by relying on only componentwise sample moments monte carlo studies indicate good power compared to hotelling s t2 and a recently proposed test by srivastava 2004 technical report university of toronto the test is applied to drug discovery data

central limit theorems for classical likelihood ratio tests for high dimensional normal distributions for random samples of size n obtained from p variate normal distributions we consider the classical likelihood ratio tests lrt for their means and covariance matrices in the high dimensional setting these test statistics have been extensively studied in multivariate analysis and their limiting distributions under the null hypothesis were proved to be chi square distributions as n goes to infinity and p remains fixed in this paper we consider the highdimensional case where both p and n go to infinity with p n y 0 1 we prove that the likelihood ratio test statistics under this assumption will converge in distribution to normal distributions with explicit means and variances we perform the simulation study to show that the likelihood ratio tests using our central limit theorems outperform those using the traditional chisquare approximations for analyzing high dimensional data

unirational threefolds with no universal codimension 2 cycle we prove that the general quartic double solid with k le 7 nodes does not admit a chow theoretic decomposition of the diagonal or equivalently has a nontrivial universal mathrm ch 0 group and the same holds if we replace in this statement chow theoretic by cohomological in particular it is not stably rational we also deduce that the general quartic double solid with seven nodes does not admit a universal codimension 2 cycle parameterized by its intermediate jacobian and even does not admit a parametrization with rationally connected fibers of its jacobian by a family of 1 cycles this finally implies that its third unramified cohomology group is not universally trivial

bias corrected diagonal discriminant rules for high dimensional classification diagonal discriminant rules have been successfully used for high dimensional classification problems but suffer from the serious drawback of biased discriminant scores in this article we propose improved diagonal discriminant rules with bias corrected discriminant scores for high dimensional classification we show that the proposed discriminant scores dominate the standard ones under the quadratic loss function analytical results on why the bias corrected rules can potentially improve the predication accuracy are also provided finally we demonstrate the improvement of the proposed rules over the original ones through extensive simulation studies and real case studies

unramified cohomology of degree 3 and noether s problem let g be a finite group and w be a faithful representation of g over c the group g acts on the field of rational functions c w the question whether the field of invariant functions c w g is purely transcendental over c goes back to emmy noether using the unramified cohomology group of degree 2 of this field as an invariant saltman gave the first examples for which c w g is not rational over c around 1986 bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group g

rost nilpotence and higher unramified cohomology abstract we develop an approach to proving the rost nilpotence principle involving higher unramified cohomology along with a decomposition of the diagonal technique we use this to prove the principle for certain threefolds over a perfect field

tests for high dimensional data based on means spatial signs and spatial ranks tests based on sample mean vectors and sample spatial signs have been studied in the recent literature for high dimensional data with the dimension larger than the sample size for suitable sequences of alternatives we show that the powers of the mean based tests and the tests based on spatial signs and ranks tend to be same as the data dimension grows to infinity for any sample size when the coordinate variables satisfy appropriate mixing conditions further their limiting powers do not depend on the heaviness of the tails of the distributions this is in striking contrast to the asymptotic results obtained in the classical multivariate setup on the other hand we show that in the presence of stronger dependence among the coordinate variables the spatial sign and rank based tests for high dimensional data can be asymptotically more powerful than the mean based tests if in addition to the data dimension the sample size also grows to infinity the sizes of some mean based tests for high dimensional data studied in the recent literature are observed to be significantly different from their nominal levels this is due to the inadequacy of the asymptotic approximations used for the distributions of those test statistics however our asymptotic approximations for the tests based on spatial signs and ranks are observed to work well when the tests are applied on a variety of simulated and real datasets

degree three unramified cohomology groups abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed subfield noether s problem asks whether k g is rational purely transcendental over k the unramified brauer group br nr c g and the unramified cohomology h nr 3 c g q z are obstructions to the rationality of c g see 14 and 5 peyre proves that if p is an odd prime number then there is a group g such that g p 12 br nr c g 0 but h nr 3 c g q z 0 thus c g is not stably c rational 12 using peyre s method we are able to find groups g with g p 9 where p is an odd prime number such that br nr c g 0 h nr 3 c g q z 0

diagonal likelihood ratio test for equality of mean vectors in high dimensional data we propose a likelihood ratio test framework for testing normal mean vectors in high dimensional data under two common scenarios the one sample test and the two sample test with equal covariance matrices we derive the test statistics under the assumption that the covariance matrices follow a diagonal matrix structure in comparison with the diagonal hotelling s tests our proposed test statistics display some interesting characteristics in particular they are a summation of the log transformed squared t statistics rather than a direct summation of those components more importantly to derive the asymptotic normality of our test statistics under the null and local alternative hypotheses we do not need the requirement that the covariance matrices follow a diagonal matrix structure as a consequence our proposed test methods are very flexible and readily applicable in practice simulation studies and a real data analysis are also carried out to demonstrate the advantages of our likelihood ratio test methods

varietes unirationnelles non rationnelles au dela de l exemple d artin et mumford keywords non rational but unirational variety grothendieck invariants rationality brauer invariants reference cma article 1989 001doi 10 1007 bf01850658 record created on 2008 12 16 modified on 2016 08 08

stably irrational hypersurfaces of small slopes we show that over any uncountable field of characteristic different from two a very general hypersurface of dimension n 2 and degree at least log 2 n 2 is not stably rational this significantly improves earlier results of kollar and totaro as a byproduct of our proof we obtain new counterexamples to the integral hodge conjecture answering a question of voisin and colliot thelene voisin

rationality of p group actions abstract let p be a prime number g a p group of order p 4 k a field with char k p if the exponent of g is p e and k contains a primitive p e th root of unity then k v g is rational purely transcendental over k where g gl v is any linear representation of g over k

a high dimensional nonparametric multivariate test for mean vector this work is concerned with testing the population mean vector of nonnormal high dimensional multivariate data several tests for high dimensional mean vector based on modifying the classical hotelling t2 test have been proposed in the literature despite their usefulness they tend to have unsatisfactory power performance for heavy tailed multivariate data which frequently arise in genomics and quantitative finance this article proposes a novel high dimensional nonparametric test for the population mean vector for a general class of multivariate distributions with the aid of new tools in modern probability theory we proved that the limiting null distribution of the proposed test is normal under mild conditions when p is substantially larger than n we further study the local power of the proposed test and compare its relative efficiency with a modified hotelling t2 test for high dimensional data an interesting finding is that the newly proposed test can have even more substantial power gain with l

cohomologie non ramifiee et conjecture de hodge entiere en nous appuyant sur la conjecture de bloch kato en k theorie de milnor nous etablissons un lien general entre le defaut de la conjecture de hodge entiere pour la cohomologie de degre 4 et le troisieme groupe de cohomologie non ramifiee a coefficients q z ceci permet de montrer que sur un solide en anglais threefold uniregle le troisieme groupe de cohomologie non ramifiee a coefficients q z s annule ce que la k theorie algebrique ne permet d obtenir que dans certains cas ceci permet a l inverse de deduire d exemples ayant leur source en k theorie que la conjecture de hodge entiere pour la cohomologie de degre 4 peut etre en defaut pour les varietes rationnellement connexes pour certaines familles a un parametre de surfaces on etablit un lien entre la conjecture de hodge entiere et l indice de la fibre generique

a two sample test for equality of means in high dimension we develop a test statistic for testing the equality of two population mean vectors in the large p small n setting such a test must surmount the rank deficiency of the sample covariance matrix which breaks down the classic hotelling t2 test the proposed procedure called the generalized component test avoids full estimation of the covariance matrix by assuming that the p components admit a logical ordering such that the dependence between components is related to their displacement the test is shown to be competitive with other recently developed methods under arma and long range dependence structures and to achieve superior power for heavy tailed data the test does not assume equality of covariance matrices between the two populations is robust to heteroscedasticity in the component variances and requires very little computation time which allows its use in settings with very large p an analysis of mitochondrial calcium concentration in mouse cardiac muscles over time and of copy number varia

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

unramified cohomology of quadrics i given a quadric x over a field f of characteristic 2 we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 galois cohomology of f to the unramified mod 2 cohomology of f x when dim x 10 and in several smaller dimensional cases applications of these results to real quadrics and to the unramified witt ring are given

smooth varieties up to a 1 homotopy and algebraic h cobordisms abstract we start to study the problem of classifying smooth proper varieties over a field k from the standpoint of a 1 homotopy theory motivated by the topological theory of surgery we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified a 1 homotopy type arithmetic considerations involving the sheaf of a 1 connected components lead us to introduce several different notions of connectedness in a 1 homotopy theory we provide concrete links between these notions connectedness of points by chains of affine lines and various rationality properties of algebraic varieties e g rational connectedness we introduce the notion of an a 1 h cobordism an algebro geometric analog of the topological notion of h cobordism and use it as a tool to produce non trivial a 1 weak equivalences of smooth proper varieties also we give explicit computations of refined a 1 homotopy invariants such as the a 1 fundamental sheaf of groups for some a 1 connected varieties we observe that the a 1 fundamental sheaf of groups plays a central yet mysterious role in the structure of a 1 h cobordisms as a consequence of these observations we completely solve the classification problem for rational smooth proper surfaces over an algebraically closed field while there exist arbitrary dimensional moduli of such surfaces there are only countably many a 1 homotopy types each uniquely determined by the isomorphism class of its a 1 fundamental sheaf of groups

a globally convergent method based on fischer burmeister operators for solving second order cone constrained variational inequality problems the karush kuhn tucker system of a second order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of fischer burmeister operators over second order cones the clarke generalized differential of the semismooth mapping is presented a modified newton method with armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem an illustrative example is given to show how the globally convergent method works

strong semismoothness of the fischer burmeister sdc and soc complementarity functions we show that the fischer burmeister complementarity functions associated to the semidefinite cone sdc and the second order cone soc respectively are strongly semismooth everywhere interestingly enough the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix

noether s problem for groups of order 32 abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed field noether s problem asks under what situations the fixed field k g will be rational purely transcendental over k theorem let g be a finite group of order 32 with exponent e if char k 2 or k is any field containing a primitive eth root of unity then k g is rational over k

noether s problem for groups of order 243 abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed field noether s problem asks under what situations the fixed field k g will be rational purely transcendental over k according to the data base of gap there are 10 isoclinism families for groups of order 243 it is known that there are precisely 3 groups g of order 243 they consist of the isoclinism family 10 such that the unramified brauer group of c g over c is non trivial thus c g is not rational over c we will prove that if 9 k then k g is rational over k for groups of order 243 other than these 3 groups except possibly for groups belonging to the isoclinism family 7

unramified brauer groups of finite simple groups of lie type a l we study the subgroup b 0 g of h 2 g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i consisting of all elements which have trivial restrictions to every abelian subgroup of g the group b 0 g serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties v g where v is a faithful complex linear representation of the group g we prove that b 0 g is trivial for finite simple groups of lie type a l

frobenius groups and retract rationality abstract let k be any field g be a finite group acting on the rational function field k x g g g by h x g x h g for any h g g define k g k x g g g g noether s problem asks whether k g is rational purely transcendental over k a weaker notion retract rationality introduced by saltman is also very useful for the study of noether s problem we prove that if g is a frobenius group with abelian frobenius kernel then k g is retract k rational for any field k satisfying some mild conditions as an application we show that for any algebraic number field k for any frobenius group g with frobenius complement isomorphic to s l 2 f 5 there is a galois extension field k over k whose galois group is isomorphic to g i e the inverse galois problem is valid for the pair g k the same result is true for any non solvable frobenius group if k 8 is a cyclic extension of k

unramified cohomology and witt groups of anisotropic pfister quadrics the unramified witt group of an anisotropic conic over a field k with char k 0 2 defined by the form 1 a b is known to be a quotient of the witt group w k of k and isomorphic to w k 1 a b ab w k we compute the unramified cohomology group h43rk c where c is the three dimensional anisotropic quadric defined by the quadratic form 1 a b ab c over k we use these computations to study the unramified witt group of c let k be a field of characteristic not two and k k be a finitely generated field extension the n th unramified cohomology group of k over k with coefficients in z 2 cf ct is denoted hnnr k k or just hnnr k when the ground field being considered is clear it is a subgroup of the galois cohomology group hn k z 2 abbreviated to hn k and is defined as hnnr k k n h6nt spec ov u2 vev k here v k is the set of all rank one discrete valuations on k that are trivial on k for v e v k ov denotes the corresponding discrete valuation ring it is well known that the group hnt spec ov j2 injects into hn k if x k is a smooth projective geometrically integral variety and k x is its function field then by abuse of terminology we refer to the groups hnrn k x as the unramified cohomology groups of x for a field f let w f be the witt group of quadratic forms over f sc for k and k k as above we can similarly define the unramified witt group of k k denoted wtr k k as the subgroup of w k defined by wnr k k n w ov vev k here w ov is the witt group of the discrete valuation ring ov which is a subgroup of w k sc chapter vi if x k is smooth projective then we refer to wnr k x as the unramified witt group of x recall that a variety x is said to be k rational if x is k birational to the projective space then the natural maps hn k hn k x resp w k w k x induce isomorphisms htn k hnrr k x resp w k wnr k x thus the unramified cohomology groups of a k rational variety resp the unramified witt received by the editors november 7 1995 1991 mathematics subject classification primary iie70 secondary 13k05 12g05

on the unramified cohomology of certain quotient varieties in this note we consider unramified cohomology with mathbb z 2 coefficients for some degree two quotient varieties and describe a method that allows one to prove the non vanishing of these groups under certain conditions we apply this method to prove a non vanishing statement in the case of kummer varieties combining this with work of colliot thelene and voisin we obtain a new type of three dimensional counterexample to the integral hodge conjecture

birational classification of fields of invariants for groups of order 128 let g be a finite group acting on the rational function field c xg g g c xg g g by cc automorphisms h xg xhgh xg xhg for any g h gg h g noether s problem asks whether the invariant field c g k xg g g gc g k xg g g g is rational i e purely transcendental over cc by fischer s theorem c g c g is rational over cc when g is a finite abelian group saltman and bogomolov respectively showed that for any prime p there exist groups g of order p9p9 and of order p6p6 such that c g c g is not rational over cc by showing the non vanishing of the unramified brauer group brnr c g 0brnr c g 0 which is an avatar of the birational invariant h3 x z torsh3 x z tors given by artin and mumford where x is a smooth projective complex variety whose function field is c g c g for p 2p 2 chu hu kang and prokhorov proved that if g is a 2 group of order 32 then c g c g is rational over cc chu hu kang and kunyavskii showed that if g is of order 64 then c g c g is rational over cc except for the groups g belonging to the two isoclinism families 13 13 with brnr c g 0brnr c g 0 and 16 16 with brnr c g c2brnr c g c2 bogomolov and bohning s theorem claims that if g1g1 and g2g2 belong to the same isoclinism family then c g1 c g1 and c g2 c g2 are stably cc isomorphic we investigate the birational classification of c g c g for groups g of order 128 with brnr c g 0brnr c g 0 moravec showed that there exist exactly 220 groups g of order 128 with brnr c g 0brnr c g 0 forming 11 isoclinism families j j we show that if g1g1 and g2g2 belong to 16 16 31 31 37 37 39 39 43 43 58 58 60 60 or 80 80 resp 106 106 or 114 114 then c g1 c g1 and c g2 c g2 are stably cc isomorphic with brnr c gi c2brnr c gi c2 explicit structures of non rational fields c g c g are given for each cases including also the case 30 30 with brnr c g c2 c2brnr c g c2 c2

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

a two sample test for high dimensional data with applications to gene set testing we proposed a two sample test for means of high dimensional data when the data dimension is much larger than the sample size the classical hotelling s t 2 test does not work for this large p small n situation the proposed test does not require explicit conditions on the relationship between the data dimension and sample size this offers much flexibility in analyzing high dimensional data an application of the proposed test is in testing significance for sets of genes which we demonstrate in an empirical study on a leukemia data set

sums of squares in function fields over henselian local fields we give upper bounds for the level and the pythagoras number of function fields over fraction fields of integral henselian excellent local rings in particular we show that the pythagoras number of mathbb r x 1 dots x n is le 2 n 1 which answers positively a question of choi dai lam and reznick