inverse numerical range and determinantal representation abstract the numerical range w a of an n n matrix a is the collection of quadratic forms z a over unit sphere 1 the inverse numerical range problem aims to find a unit vector which corresponds to a given point z of the numerical range w a given a matrix a the helton vinnikov theorem produces a symmetric matrix s so that a and s have the same numerical range w a w s in this paper we investigate the inverse numerical range problem for the boundary points and points on the boundary generating curve of the numerical range in place of the construction of unit vectors satisfying z a we express the kernel vector function of the linear pencil x s y s z i n as a function on the abel jacobi variety of the associated elliptic curve of a the kernel function plays a key role for the inverse numerical problem we perform this process when s is a generic 3 3 symmetric matrix

convergence of conditional expectations and strong laws of large numbers for multivalued random variables on etablit des lemmes de fatou et des theoremes de convergence de lebesgue pour des esperances conditionnelles multivaluees de variables aleatoires a valeurs dans des sous ensembles fermes d un espace de banach separable

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

obstructions to determinantal representability there has recently been ample interest in the question of which sets can be represented by linear matrix inequalities lmis a necessary condition is that the set is rigidly convex and it has been conjectured that rigid convexity is also sufficient to this end helton and vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices we disprove this conjecture by relating the question of finding lmi representations to the problem of determining whether a polymatroid is representable over the complex numbers we find a real zero polynomial such that no power of it admits a determinantal representation the proof uses recent results of wagner and wei on matroids with the half plane property and the polymatroids associated to hyperbolic polynomials introduced by gurvits

measurability and integrability of the weak upper limit of a sequence of multifunctions we provide several properties of the weak upper limit of a sequence of subsets of a separable banach space such as a criterion of non vacuity of closedness etc we also examine the measurability of the multifunction defined as the weak upper limit of a sequence of multifunctions at last applications to the existence of a measurable and bochner integrable selector for this multifunction are presented

lower closure problems with weak convergence conditions in a new perspective a transparent approach is taken towards the class of lower closure problems with weak convergence conditions for the derivatives a deparametrization procedure is formulated for the abstract lower closure problem of this class as well as for its variant in a control problem of lagrange type it is shown that if one follows this procedure the solution of the lower closure problem merely lies in proving that a certain modified lagrangian is a normal integrand a lower closure result is obtained that generalizes in itself all comparable results of 2a 6e f 7a 21 for control problems of lagrange type with uniform boundedness conditions on the controls a special approximate lagrangian can be formulated by which the deparametrization procedure yields results that are superior to those obtained previously 2b 7b c an important novelty in deriving the measurability properties of lagrangians is also introduced here it consists of the employment of the kunugui novikov stchegolkov proj

hyperbolic polynomials and interior point methods for convex programming hyperbolic polynomials have their origins in partial differential equations we show in this paper that they have applications in interior point methods for convex programming each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone we give an explicit representation of this cone in terms of polynomial inequalities the function fx log px is a logarithmically homogeneous self concordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p the function fx possesses striking additional properties that are useful in designing long step interior point methods for example we show that the long step primal potential reduction methods of nesterov and todd and the surface following methods of nesterov and nemirovskii extend to hyperbolic barrier functions we also show that there exists a hyperbolic barrier function on every homogeneous cone

on the minimum problem for a class of noncoercive nonconvex functionals we are concerned with the problem of existence of solutions to the variational problem min left int 0 r g t v t dt v in ac 0 r v r 0 right with only one fixed endpoint prescribed the map g colon 0 r times mathbb r to overline mathbb r is a normal integrand for which neither convexity nor superlinear growth conditions are assumed as an application we give an existence result for the radially symmetric variational problem min u in wuu b r int b r f pt mod x mod nabla u x a x u x dx where b r is the ball of mathbb r n centered at the origin and with radius r 0 the map f colon 0 r times 0 infty to overline mathbb r is a normal integrand and a in l 0 r again neither convexity nor superlinear growth conditions are made on f these kinds of problems with nonconvex lagrangians with respect to nabla u arise in different fields of mathematical physics such as optimal design and nonlinear elasticity

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

new existence results for optimal controls in the absence of convexity the importance of extremality a new approach to existence without convexity is presented instead of applying a weierstrass tonelli type existence result to a convexified i e relaxed form of the control problem which is standard it uses a new extremum principle this principle guarantees the existence of an optimal relaxed control function that is a convex combination of relaxations of ordinary control functions the same linear relationship is maintained for the corresponding trajectories and this is of much benefit in the final phase which consists of a deconvexification by a global subgradient argument and lyapunov s theorem the extremum principle is based on extreme point features of the relaxed control problem it can be stated in an abstract form and extends the classical extremum principle of bauer the new approach applies both to optimal control problems of suitable type dynamics linear in the state variable and to abstract variational problems without dynamics well known in economics with a different interpretation it can also give bang bang existence results several new applications serve to illustrate the power of the approach

convergence of conditional expectations for unbounded random sets integrands and integral functionals given a sequence of unbounded convex random sets we study under which conditions fatou s lemma for the weak upper limit of their conditional expectations holds we also give multivalued versions of dominated and monotone convergence theorems and we discuss the special case of the integral finally applications to epigraphic convergence of integrands and to mosco convergence of certain integral functionals are provided

indefinite hardy spaces on finite bordered riemann surfaces abstract it is well known that one may study hardy spaces with the domain a finite bordered riemann surface rather than the unit disk however for domains with more than one boundary component it is natural to consider besides the usual positive definite inner product on h 2 indefinite inner products obtained by picking up different signs or in the vector valued case different signature matrices when integrating over different components in this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a krein space furthermore each holomorphic mapping of the finite bordered riemann surface onto the unit disk which maps boundary to boundary determines an explicit isometric isomorphism between this space and a usual vector valued hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix as is usual when studying hardy spaces on a multiply connected domain the elements of the space are sections of a vector bundle rather than functions the main point is to construct an appropriate extension of this bundle to the double of the finite bordered riemann surface and to use cauchy kernels for certain vector bundles on a compact riemann surface

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

the resultant on compact riemann surfaces we introduce a notion of the resultant of two meromorphic functions on a compact riemann surface and demonstrate its usefulness in several respects for example we exhibit several integral formulas for the resultant relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact riemann surface as a particular application the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the schottky double of the domain

complete description of determinantal representations of smooth irreducible curves abstract determinantal representations of algebraic curves are interesting in themselves and their classification is equivalent to the simultaneous classification of triples of matrices we present a complete description of determinantal representations of smooth irreducible curves over any algebraically closed field we used the notion of the class of divisors of the vector bundle corresponding to a determinantal representation we prove that two determinantal representations of a smooth curve f are equivalent if and only if the classes of divisors of the corresponding vector bundles coincide we give a precise characterization of those classes of divisors that arise from vector bundles corresponding to determinantal representations of f then we obtain a parametrization of determinantal representations of f up to equivalence by the points of the jacobian variety of f not on some exceptional subvariety in particular it follows that any smooth curve of order 3 or greater possesses an infinite number of nonequivalent determinantal representations we also specialize our results to symmetrical and self adjoint representations

the cayley hamilton theorem and inverse problems for multiparameter systems abstract we review some of the current research in multiparameter spectral theory we prove a version of the cayley hamilton theorem for multiparameter systems and list a few inverse problems for such systems some consequences of results on determinantal representations proved by dixon dickson and vinnikov for the inverse problems are discussed

how to count zeroes of polynomials on quadrature domains using the bezout matrix abstract classically the bezout matrix or simply bezoutian of two polynomials is used to locate the roots of the polynomial and in particular test for stability in this paper we develop the theory of bezoutians on real riemann surfaces of dividing type the main result connects the signature of the bezoutian of two real meromorphic functions to the topological data of their quotient which can be seen as the generalization of the classical cauchy index as an application we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the bezoutian we provide examples of our method in the case of simply connected quadrature domains

representations of lie algebras by non skewselfadjoint operators in hilbert space we study non skewselfadjoint representations of a finite dimensional real lie algebra gg to this end we embed a non skewselfadjoint representation of gg into a more complicated structure that we call a gg operator vessel and that is associated to an overdetermined linear conservative input state output system on the corresponding simply connected lie group gg we develop the frequency domain theory of the system in terms of representations of gg and introduce the joint characteristic function of a gg operator vessel which is the analogue of the classical notion of the characteristic function of a single non selfadjoint operator as the first non commutative example we apply the theory to the lie algebra of the ax bax b group the group of affine transformations of the line

on ws convergence of product measures a number of fundamental results centered around extensions of prohorov s theorem is proven for the ws topology for measures on a product space these results contribute to the foundations of stochastic decision theory they also subsume the principal results of young measure theory which only considers product measures with a fixed common marginal specializations yield the criterion for relative ws compactness of scha l 1975 the refined characterizations of ws convergence of galda ano and truffert 1997 1998 and a new version of fatou s lemma in several dimensions in a separate nonsequential development a generalization is given of the relative ws compactness criterion of jacod and ma min 1981 new applications are given to the existence of optimal equilibrium distributions over player action pairs in game theory and the existence of most optimistic scenarios in stochastic decision theory

existence without explicit compactness in stochastic dynamic programming a new kind of existence result for stochastic dynamic programming without explicit compactness conditions is obtained by working with a new coercivity condition for the reward functions

comment on an existence result for a noncoercive nonconvex variational problem an essential improvement is given of a recent existence result of crasta g crasta siam j control optim 38 1999 pp 237 253 for a nonconvex noncoercive variational problem whose integrand does not depend on the state variable this is shown to follow by the methods based on fatou s lemma in several dimensions of e j balder j math anal appl 101 1984 pp 527 539 the associated euler lagrange inclusions follow from well known optimality conditions for such problems v m alekseev v m tichimirov and s v fomin optimal control consultants bureau new york 1987 v i arkin and v l levin uspekhi mat nauk 27 1972 pp 21 77

a representation theorem for aumann integrals in recent years the study of set valued functions has been developed extensively by many authors with applications to mathematical economics and control theory see refs 5 11 15 161 in those papers three approaches can be distinguished according to whether the range space values of setvalued functions is z a banach space or a locally convex topological space the purpose of this paper is to establish properties of aumann s integrals of set valued functions f t 2 whose values are nonempty subsets of a real separable reflexive banach space x and to continue the work due to aumann 2 and datko 7 81 while previous analysis has always treated the case of special finite nonatomic measure spaces we focus here on the case of general a finite nonatomic measure spaces in this last situation moreover the analogous results we establish hold under less stringent hypotheses more precisely all through the paper we consider a measure space t c p where p is supposed to be positive nonatomic and u finite and we give the following statements

optimal detection of heterogeneous and heteroscedastic mixtures the problem of detecting heterogeneous and heteroscedastic gaussian mixtures is considered the focus is on how the parameters of heterogeneity heteroscedasticity and proportion of non null component influence the difficulty of the problem we establish an explicit detection boundary which separates the detectable region where the likelihood ratio test is shown to reliably detect the presence of non null effect from the undetectable region where no method can do so in particular the results show that the detection boundary changes dramatically when the proportion of nonnull component shifts from the sparse regime to the dense regime furthermore it is shown that the higher criticism test which does not require the specific information of model parameters is optimally adaptive to the unknown degrees of heterogeneity and heteroscedasticity in both the sparse and dense cases

livsic type determinantal representations and hyperbolicity abstract hyperbolic homogeneous polynomials with real coefficients i e hyperbolic real projective hypersurfaces and their determinantal representations play a key role in the emerging field of convex algebraic geometry in this paper we consider a natural notion of hyperbolicity for a real subvariety x p d of an arbitrary codimension l with respect to a real l 1 dimensional linear subspace v p d and study its basic properties we also consider a class of determinantal representations that we call livsic type and a nice subclass of these that we call very reasonable much like in the case of hypersurfaces l 1 the existence of a definite hermitian very reasonable livsic type determinantal representation implies hyperbolicity we show that every curve admits a very reasonable livsic type determinantal representation our basic tools are cauchy kernels for line bundles and the notion of the bezoutian for two meromorphic functions on a compact riemann surface that we introduce we then proceed to show that every real curve in p d hyperbolic with respect to some real d 2 dimensional linear subspace admits a definite hermitian or even definite real symmetric very reasonable livsic type determinantal representation

on linearizations of the quadratic two parameter eigenvalue problems abstract we present several transformations that can be used to solve the quadratic two parameter eigenvalue problem qmep by formulating an associated linear multiparameter eigenvalue problem two of these transformations are generalizations of the well known linearization of the quadratic eigenvalue problem and linearize the qmep as a singular two parameter eigenvalue problem the third replaces all nonlinear terms by new variables and adds new equations for their relations the qmep is thus transformed into a nonsingular five parameter eigenvalue problem the advantage of these transformations is that they enable one to solve the qmep using existing numerical methods for multiparameter eigenvalue problems we also consider several special cases of the qmep where some matrix coefficients are zero

real separated algebraic curves quadrature domains ahlfors type functions and operator theory abstract the aim of this paper is to inter relate several algebraic and analytic objects such as real type algebraic curves quadrature domains functions on them and rational matrix functions with special properties and some objects from operator theory such as vector toeplitz operators and subnormal operators our tools come from operator theory but some of our results have purely algebraic formulation we make use of xia s theory of subnormal operators and of the previous results by the author in this direction we also correct in section 5 some inaccuracies in the works of d v yakubovich subnormal operators of finite type i xia s model and real algebraic curves in c 2 rev mat iberoamericana 14 1998 95 115 d v yakubovich subnormal operators of finite type ii structure theorems rev mat iberoamericana 14 1998 623 681 by the author

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

optimal classification in sparse gaussian graphic model consider a two class classification problem where the number of features is much larger than the sample size the features are masked by gaussian noise with mean zero and covariance matrix sigma where the precision matrix omega sigma 1 is unknown but is presumably sparse the useful features also unknown are sparse and each contributes weakly i e rare and weak to the classification decision by obtaining a reasonably good estimate of omega we formulate the setting as a linear regression model we propose a two stage classification method where we first select features by the method of innovated thresholding it and then use the retained features and fisher s lda for classification in this approach a crucial problem is how to set the threshold of it we approach this problem by adapting the recent innovation of higher criticism thresholding hct we find that when useful features are rare and weak the limiting behavior of hct is essentially just as good as the limiting behavior of ideal threshold the threshold one would choose if the underlying distribution of the signals is known if only somewhat surprisingly when omega is sufficiently sparse its off diagonal coordinates usually do not have a major influence over the classification decision compared to recent work in the case where omega is the identity matrix proc natl acad sci usa 105 2008 14790 14795 philos trans r soc lond ser a math phys eng sci 367 2009 4449 4470 the current setting is much more general which needs a new approach and much more sophisticated analysis one key component of the analysis is the intimate relationship between hct and fisher s separation another key component is the tight large deviation bounds for empirical processes for data with unconventional correlation structures where graph theory on vertex coloring plays an important role

on seminormality of integral functionals and their integrands we present a definition of seminormality which extends classical notions due to tonelli fondamenti di calcolo delle variazioni 1921 mcshane ann scuola norm sup pisa 2 3 1934 pp 181 211 287 315 and cesari traps amer math soc 124 1966 pp 369 412 j optim theory appl 6 1970 pp 114 137 siam j control optim 9 1971 pp 287 315 and applies to both integrands seminormality in the small and their integral functionals seminormality in the large our main result states that under very general hypotheses seminormality in the small and large are equivalent by introducing a notion called nagumo tightness we show that the usual sufficient conditions for the lower semicontinuity of an integral functional also imply a form of seminormality of the integral functional necessary conditions for the lower semicontinuity of an integral functional can also be obtained from our results

a general fatou lemma a general fatou lemma is established for a sequence of gelfand integrable functions from a vector loeb space to the dual of a separable banach space or with a weaker assumption on the sequence a banach lattice a corollary sharpens previous results in the finite dimensional setting even for the case of scalar measures counterexamples are presented to show that the results obtained here are sharp in various aspects applications include systematic generalizations of the distribution of correspondences from the case of scalar loeb spaces to the case of vector loeb spaces and a proof of the existence of a pure strategy equilibrium in games with private and public information and with compact metric action spaces

a framework for inference using goodness of fit tests based on ensemble of phi divergences in this paper we study the inferential use of goodness of fit tests in a non parametric setting the utility of such tests will be demonstrated for the test case of spectrum sensing applications in cognitive radios we provide the first comprehensive framework for decision fusion of an ensemble of goodness of fit testing procedures through an ensemble goodness of fit test also we introduce a generalized family of functionals and kernels called divergences which allow us to formulate goodness of fit tests that are parameterized by a single parameter the performance of these tests is simulated under gaussian and non gaussian noise in a mimo setting we show that under uncertainty in the noise statistics or non gaussianity in the noise the performance of non parametric tests in general and phi divergence based goodness of fit tests in particular is significantly superior to that of the energy detector with reduced implementation complexity in particular the false alarm rates of our proposed tests is maintained at a fixed level over a wide variation in the channel noise distributions additionally we describe a collaborative spatially separated version of the test for robust combining of tests in a distributed spectrum sensing setting and quantify the significant collaboration gains achieved

on the correlation functions of the vector bundle generalization of the bc system it is shown that the determinants of the correlation functions of the generalized bc system introduced recently are given as pullbacks of the non abelian theta divisor

global power functions of goodness of fit tests usseldorf it is shown that the global power function of any nonparametric test is flat on balls of alternatives except for alternatives coming from a finite dimensional subspace the present benchmark is here the upper one sided or two sided envelope power function every choice of a test fixes a priori a finite dimensional region with high power it turns out that also the level points are far away fromthe corresponding neym an pearson test level points except for a finite number of orthogonal directions of alternatives for certain submodels the result is independent of the underlying sample size in the last section the statistical consequences and special goodness of fit tests are discussed 1 introduction omnibus tests are commonly used if the specific structure of certain nonparametric alternatives is unknown among other justifications it turns out that they typically are consistent against fixed alternatives and n consistent under sequences of local alternatives of sample size n f or these reasons people often trust in goodness of fit tests and these are frequently applied to data of finite sample size on the other hand every asymptotic approximation should be understood as an approximation of the underlying finite sample case thus the statistician likes to distinguish and to compare the power of different competing tests the present paper offers a concept for the comparison and justification of different tests by their power functions and level points it is shown that under certain circumstances every test has a preference for a finite dimensional space of alternatives apart fromthis space the power function is almost flat on balls of alternatives there exists no test which pays equal attention to an infinite number of orthogonal alternatives the results do not only hold for asymptotic models but they also hold for concrete alternatives on the real line at finite sample size and their level points uniformly for the sample size the results are not surprising every statistician knows that it is impossible to separate an infinite sequence of different parameters simultaneously if only a finite number of observations is available the conclusions of the results are two fold 1 the statistician should analyze the goodness of fit tests of his computer package in order to get some knowledge and an impression about their preferences

new sequential compactness results for spaces of scalarly integrable functions abstract a new general relative sequential compactness criterion for scalarly integrable functions is presented the novel feature of this result is the fact that it deals with a strong form of pointwise cesaro convergence almost everywhere which is stronger than the usual types of weak convergence applications of the main result include extensions of a classical weak compactness criterion for abstract l1 spaces and a recent relative sequential compactness criterion of prohorov type for transition probabilites the main result can also be seen as an abstract version of a famous theorem of komlos which is also essential for its proof

estimating the proportion of false null hypotheses among a large number of independently tested hypotheses we consider the problem of estimating the number of false null hypotheses among a very large number of independently tested hypotheses focusing on the situation in which the proportion of false null hypotheses is very small we propose a family of methods for establishing lower 100 1 a confidence bounds for this proportion based on the empirical distribution of the p values of the tests methods in this family are then compared in terms of ability to consistently estimate the proportion by letting 0 as the number of hypothesis tests increases and the proportion decreases this work is motivated by a signal detection problem that occurs in astronomy

robust solutions of optimization problems affected by uncertain probabilities in this paper we focus on robust linear optimization problems with uncertainty regions defined by divergences for example chi squared hellinger kullback leibler we show how uncertainty regions based on divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector such problems frequently occur in for example optimization problems in inventory control or finance that involve terms containing moments of random variables expected utility etc we show that the robust counterpart of a linear optimization problem with divergence uncertainty is tractable for most of the choices of typically considered in the literature we extend the results to problems that are nonlinear in the optimization variables several applications including an asset pricing example and a numerical multi item newsvendor example illustrate the relevance of the proposed approach this paper was accepted by gerard p cachon optimization

properties of higher criticism under strong dependence the problem of signal detection using sparse faint information is closely related to a variety of contemporary statistical problems including the control of false discovery rate and classification using very high dimensional data each problem can be solved by conducting a large number of simultaneous hypothesis tests the properties of which are readily accessed under the assumption of independence in this paper we address the case of dependent data in the context of higher criticism methods for signal detection short range dependence has no first order impact on performance but the situation changes dramatically under strong dependence there although higher criticism can continue to perform well it can be bettered using methods based on differences of signal values or on the maximum of the data the relatively inferior performance of higher criticism in such cases can be explained in terms of the fact that under strong dependence the higher criticism statistic behaves as though the data were partitioned into very large blocks with all but a single representative of each block being eliminated from the dataset

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

an algebraic geometry study of the b c system with arbitrary twist fields and arbitrary statistics we present an analysis of the generalb c system including the system on a compact riemann surface of arbitrary genusg 0 by postulating that its correlation functions should only have the singularities imposed by the operator product expansion ope of the system studying a very in fact optimally general form of theb c system we prove rigorously that the standard practice of eliminating zero modes and even the standard lagrangian follow from the analyticity structure dictated by the ope alone we extend the analysis to consider the most general case of the presence of twist e g spin fields we then determine all the possible correlation functions of theb c system with statistics unspecified compatible with the ope on imposing fermi and bose statistics we obtain the correlation functions of the fermionicb c and systems respectively

elementary transformations of pfaffian representations of plane curves abstract let c be a smooth curve in p 2 given by an equation f 0 of degree d in this paper we consider elementary transformations of linear pfaffian representations of c elementary transformations can be interpreted as actions on a rank 2 vector bundle on c with canonical determinant and no sections which corresponds to the cokernel of a pfaffian representation every two pfaffian representations of c can be bridged by a finite sequence of elementary transformations pfaffian representations and elementary transformations are constructed explicitly for a smooth quartic applications to aronhold bundles and theta characteristics are given

on the existence of nash equilibrium in bayesian games we furnish conditions on the primitives of a bayesian game that guarantee the existence of a bayes nash equilibrium by allowing for payoff discontinuities in actions we cover various applications that cannot be handled by extant results

the bc system of higher rank revisited the bc system of higher rank introduced recently is examined further it is shown that the correlation functions are connected with certain non abelian functions and it is discussed how quasi determinants arise

finite dimensional de branges spaces on riemann surfaces abstract we study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real riemann surface these spaces are analogues of the spaces introduced by l de branges when the riemann sphere is replaced by a compact real riemann surface of a higher genus in de branges theory an important role is played by resolvent like difference quotient operators r here we introduce generalized difference quotient operators r y for any non constant meromorphic function y on the riemann surface the spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators r y 1 i and r y 2 2 where y 1 and y 2 generate the field of meromorphic functions on the riemann surface which satisfy a supplementary identity analogous to the de branges identity for difference quotients just as the classical de branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint or nonunitary operator the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks

the compactness of a policy space in dynamic programming via an extension theorem for carathe odory functions the background for this paper is a dynamic programming model with a borel state space and compact action sets a new simple proof of the compactness of a space of measures corresponding to randomized stationary policies is given in a topology induced by caratheodory functions on the state action space for this purpose an extension theorem for caratheodory functions is proved which is a new version of the tietze s extension theorem for continuous functions

relaxed inf compactness for variational problems by hilbert cube compactification abstract inf compactness of the objective functional in an abstract variational problem where the dynamical system consists of a linear functional integral equation is proven by suitably topologizing the space of relaxed control functions the topology is obtained quite naturally from a hilbert cube compactification of the space of control points under additional convexity assumptions a general existence result implying that of a d ioffe dokl akad nauk sssr 205 1972 277 280 soviet math dokl 13 1972 919 923 is proven

existence results without convexity conditions for general problems of optimal control with singular components this note presents a new quick approach to existence results without convexity conditions for optimal control problems with singular components in the sense of e i mcshane siam j control 5 1967 438485 starting from the resolvent kernel representation of the solutions of a linear integral equation a version of fatou s lemma in several dimensions is shown to lead directly to a compactness result for the attainable set and an existence result for a mayer problem these results subsume those of l w neustadt j math anal appl 7 1963 110 l 17 c olech j dlfirential equations 2 1966 74 iol m q jacobs mathematical theory of control pp 46 53 academic press 1967 l cesari siam j control 12 1974 319 331 and t s angel1 j optim theory appl 19 1976 63 79

an existence result for optimal economic growth problems abstract an existence result for optimal control problems of lagrange type with unbounded time domain is derived very directly from a corresponding result for problems with bounded time domain this subsumes the main existence result of r f baum j optim theory appl 19 1976 89 116 and has the existence results for optimal economic growth problems of s i takekuma j math econom 7 1980 193 208 and m j p magill econometrica 49 1981 679 711 j math anal appl 82 1981 66 74 as simple corollaries in addition a new notion of uniform integrability is used which coincides with the classical notion if the time domain is bounded

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

a unifying note on fatou s lemma in several dimensions a general version of fatou s lemma in several dimensions is presented it subsumes the fatou lemmas given by schmeidler schmeidler d 1970 fatou s lemma in several dimensions proc amer math soc 24 300 306 hildenbrand hildenbrand w 1974 core and equilibria of a large economy princeton university press princeton cesari suryanarayana cesari l m b suryanarayana 1978 an existence theorem for pareto problems nonlinear anal 2 225 233 and artstein artstein z 1979 a note on fatou s lemma in several dimensions j math econom 6 277 282 also it is equivalent to an abstract variational existence result that extends and generalizes results by aumann perles aumann r j m perles 1965 a variational problem arising in economics j math anal appl 11 488 503 berliocchi lasry berliocchi h j m lasry 1973 integrands normales et mesures parametrees en calcul des variations bull soc math france101 129 184 artstein artstein z 1974 on a variational problem j math anal appl 45 404 415 and balder balder e j 1979 on a useful compactification for optimal control problems j math anal appl 72 391 398 in several respects

on the characteristic polynomials of multiparameter pencils abstract in this paper we study the characteristic polynomial of multiparameter pencil z 1 a 1 z 2 a 2 z s a s the main theorem states that a unitary representation of a finitely generated group contains a one dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor it follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible the result is of kin to the dedekind and frobenius theorem on finite group determinant

on the quadratic two parameter eigenvalue problem and its linearization we introduce the quadratic two parameter eigenvalue problem and linearize it as a singular two parameter eigenvalue problem this together with an example from model updating shows the need for numerical methods for singular two parameter eigenvalue problems and for a better understanding of such problems there are various numerical methods for two parameter eigenvalue problems but only few for nonsingular ones we present a method that can be applied to singular two parameter eigenvalue problems including the linearization of the quadratic two parameter eigenvalue problem it is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils

nonparametric likelihood confidence bands for a distribution function abstract berk and jones described a nonparametric likelihood test of uniformity with greater asymptotic bahadur efficiency than any weighted kolmogorov smirnov test at any alternative to u 0 1 we invert this test to form confidence bands for a distribution function using noe s recursion nonparametric likelihood bands are narrower in the tails and wider in the center than kolmogorov smirnov bands and are asymmetric about the empirical cumulative distribution function this article describes how to convert a confidence level into a likelihood threshold and how to use the threshold to compute bands simple computation saving approximations to the threshold are given for confidence levels 95 and 99 and all sample sizes up to 1 000 these yield coverage between the nominal and 01 over the nominal the likelihood bands are illustrated on some galaxy velocity data and are shown to improve power over kolmogorov smirnov bands on some examples with n 20

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

resultants and chow forms via exterior syzygies let w be a vector space of dimension n 1 over a field k the chow divisor of a k dimensional variety x in p p w is the hypersurface in the grassmannian gk 1 of planes of codimension k 1 in p whose points over the algebraic closure of k are the planes that meet x the chow form of x is the defining equation of the chow divisor for example the resultant of k 1 forms of degree e in k 1 variables is the chow form of p embedded by the e th veronese mapping in p with n k e k 1 more generally the chow divisor of a k cycle i ni vi on projective space is defined to be i nidi where di is the chow divisor of vi the chow divisor of a sheaf f with k dimensional support is the chow divisor of the associated k cycle of f in this paper we will give a new expression for the chow divisor and apply it to give explicit formulas in many new cases starting with a sheaf f on p we use exterior algebra methods to define a canonical and effectively computable chow complex of f on each grassmannian of planes in p if f has k dimensional support we show that the chow form of f is the determinant of the chow complex of f on the grassmannian of planes of codimension k 1 the beilinson monad of f beilinson 1978 is the chow complex of f on the grassmannian of 0 planes that is on p itself in particular we are able to give explicit determinantal and pfaffian formulas for resultants in some cases where no polynomial formulas were known for example the horrocks mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4 6 or 8 in five variables the easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables the chow form of the veronese surface in p using the tangent bundle of p conclude that it can be written in bezout form described below as the pfaffian of the matrix 0 245 345 135 045 035 145 235

interpolation problems for holomorphic functions abstract in this note we discuss the nevanlinna pick problem corona problem and caratheodory fejer problem for bounded holomorphic functions

goodness of fit tests via phi divergences a unified family of goodness of fit tests based on o divergences is introduced and studied the new family of test statistics s n s includes both the supremum version of the anderson darling statistic and the test statistic of berk and jones z wahrsch verw gebiete 47 1979 47 59 as special cases s 2 and s 1 resp we also introduce integral versions of the new statistics we show that the asymptotic null distribution theory of berk and jones z wahrsch verw gebiete 47 1979 47 59 and wellner and koltchinskii high dimensional probability iii 2003 321 332 birkhauser basel for the berk jones statistic applies to the whole family of statistics s n s with s 1 2 on the side of power behavior we study the test statistics under fixed alternatives and give extensions of the poisson boundary phenomena noted by berk and jones for their statistic we also extend the results of donoho and jin ann statist 32 2004 962 994 by showing that all our new tests for s 1 2 have the same optimal detection boundary for normal shift mixture alternatives as tukey s higher criticism statistic and the berk jones statistic

generalized equilibrium results for games with incomplete information milgrom and weber milgrom p r weber r j 1985 distributional strategies for games with incomplete information math oper res 10 619 632 gave an existence result for a nash equilibrium in a game with incomplete information using their notion of a distributional strategy here we obtain a substantial improvement of their existence result in terms of the more traditional concept of a behavioral strategy this improvement is reached very naturally as an application of a theory of weak convergence for transition probabilities which is recapitulated extensively in this paper also a new result on the weak convergence of product transition probabilities is included

distributions and power of optimal signal detection statistics in finite case for detecting weak and sparse signals by a set of n input p values the higher criticism hc type statistics the berk jones b j type statistics and the phi divergence statistics have the equivalent asymptotic optimality as n goes to infinity however they can have significantly different performance in practical data analysis where n is always finite and even very small to address this problem in a broader context this paper introduces a general family of goodness of fit statistics called the ggof which unifies a broad signal detection statistics including these optimal ones efficient and accurate analytical calculations for the distributions of the ggof statistics are provided under arbitrary i i d continuous models of the null and the alternative hypotheses based on that a systematic power study reveals that in finite case the number of signals is often more relevant than the signal proportion the hc and the reverse hc have advantages for relatively sparser and denser signals respectively while the b j is more robust a general framework is given to apply the ggof into data analysis based on the generalized linear models an application to the snp set based genome wide association study gwas for crohn s disease shows that these optimal statistics have a good potential for detecting novel disease genes with weak snp effects the calculations have been implemented into an r package settest and published on the cran

optimal detection of sparse mixtures against a given null distribution detection of sparse signals arises in a wide range of modern scientific studies the focus so far has been mainly on gaussian mixture models in this paper we consider the detection problem under a general sparse mixture model and obtain explicit expressions for the detection boundary under mild regularity conditions in addition for gaussian null hypothesis we establish the adaptive optimality of the higher criticism procedure for all sparse mixtures satisfying the same conditions in particular the general results obtained in this paper recover and extend in a unified manner the previously known results on sparse detection far beyond the conventional gaussian model and other exponential families

fatou s lemma for multifunctions with unbounded values for multifunctions having unbounded sets as values we give fatou type inclusions for the kuratowski limes superior in finite as well as infinite dimensions these are derived from similar known fatou type inequalities for single valued multifunctions i e ordinary functions that is from balder s unifying fatou lemma in case the image set is finite dimensional and from an update of related results by balder in the infinite dimensional case for this extension from the single valued to the multivalued situation a lemma due to hess used to prove earlier fatou type inclusions is of critical importance also an asymptotic correction term introduced here plays an important role the two man results thus obtained subsume and extend an entire class of comparable fatou lemmas

elementary transformations of determinantal representations of algebraic curves abstract we study elementary transformations first introduced by livsic and kravitsky in an operator theoretic context of determinantal representations of algebraic curves we consider determinantal representations of a smooth irreducible curve f over an algebraically closed field when regarded as acting on the corresponding vector bundle the elementary transformations are a matrix generalization of scalar linear fractional transformations they add to the class of divisors of the vector bundle a single zero and a single pole we show that given a determinantal representation of f we can build all the nonequivalent determinantal representations of f by applying to the given representation finite sequences of at most g elementary transformations where g is the genus of f

geometric aspects of multiparameter spectral theory the paper contains a geometric description of the dimension of the total root subspace of a regular multiparameter system in terms of the intersection multiplicities of its determinantal hypersurfaces the new definition of regularity used is proved to restrict to the familiar definition in the linear case a decomposability problem is also considered it is shown that the joint root subspace of a multiparameter system may be decomposable even when the root subspace of each member is indecomposable

empirical likelihood ratio test with distribution function constraints in this work we study non parametric hypothesis testing problem with distribution function constraints the empirical likelihood ratio test has been widely used in testing problems with moment in equality constraints however some detection problems cannot be described using moment in equalities we propose a distribution function constraint along with an empirical likelihood ratio test this detector is applicable to a wide variety of robust parametric non parametric detection problems since the distribution function constraints provide a more exact description of the null hypothesis the test outperforms the empirical likelihood ratio test with moment constraints as well as many popular goodness of fit tests such as the robust kolmogorov smirnov test and the cramer von mises test examples from communication systems with real world noise samples are provided to show their performance specifically the proposed test significantly outperforms the robust kolmogorov smirnov test and the cramer von mises test when the null hypothesis is nested in the alternative hypothesis the same example is repeated when we assume no noise uncertainty by doing so we are able to claim that in our case it is necessary to include uncertainty in noise distribution additionally the asymptotic optimality of the proposed test is provided

de branges spaces on compact riemann surfaces and a beurling lax type theorem abstract using the notion of commutative operator vessels this work investigates de branges rovnyak spaces whose elements are sections of a line bundle of multiplicative half order differentials on a compact real riemann surface as a special case we obtain a beurling lax type theorem in the setting of the corresponding hardy space on a finite bordered riemann surface

lower semicontinuity of integral functionals with nonconvex integrands by relaxation compactification a new approach to the lower semicontinuity of integral functionals is presented by a topological embedding of the control and state spaces in the hilbert cube and a simultaneous relaxation of the control functions a powerful approach emerges whose main features include i a generalized convexity condition is imposed upon the integrand of which the classical convexity condition is a special case ii in the embedded setting the integrand can be supposed lipschitz continuous in control and state arguments without loss of generality iii convergence in measure of the trajectories metamorphoses into l 1 norm convergence in the embedded setting

partial correlation screening for estimating large precision matrices with applications to classification given n samples x1 x2 xn from n 0 we are interested in estimating the p p precision matrix 1 we assume is sparse in that each row has relatively few nonzeros we propose partial correlation screening pcs as a new row by row approach to estimate the i th row of 1 i p pcs uses a screen step and a clean step in the

determinantal representations of invariant hyperbolic plane curves abstract we study hyperbolic polynomials with nice symmetry and express them as the determinant of a hermitian matrix with special structure the goal of this paper is to answer a question posed by chien and nakazato in 2015 3 by properly modifying a determinantal representation construction of dixon 1902 5 we show for every hyperbolic polynomial of degree n invariant under the cyclic group of order n there exists a determinantal representation admitted via some cyclic weighted shift matrix moreover if the polynomial is invariant under the action of the dihedral group of order n the associated cyclic weighted shift matrix is unitarily equivalent to one with real entries

symmetric determinantal representation of polynomials abstract we give an elementary proof only using linear algebra of a result due to helton mccullough and vinnikov which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil we give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2

a general approach to lower semicontinuity and lower closure in optimal control theory a self contained approach to lower semicontinuity and lower closure evolves from an extension of relaxed control theory which is based on a central relative weak compactness criterion called tightness and relaxation in all but one variable two lower closure results for outer integral functionals with variable abstract time domain are developed the first of these has a convexity condition for the integrand and generalizes all similar results in the literature the second lower closure result is of a new kind among other things it implies a quite general version of fatou s lemma in several dimensions