scalar curvature of a metric with unit volume the problem of finding riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of kazdan and warner in 1970 s 10 and references cited in it in this paper we shall consider the same problem under a constraint on the volume for this purpose it is useful to introduce an invariant p m of a smooth closed manifold m which will be called the yamabe number of m defined as the supremum of m c of all conformal classes c of riemannian metrics on m

on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

computing nonnegative rank factorizations abstract the existence of nonnegative generalized inverses in terms of nonnegative rank factorizations is considered an algorithm is presented which computes a nonnegative rank factorization of a nonnegative matrix when a nonnegative 1 inverse exists

solving the kohn laplacian on asymptotically flat cr manifolds of dimension 3 abstract let x t 1 0 x be a compact orientable cr embeddable three dimensional strongly pseudoconvex cr manifold where t 1 0 x is a cr structure on x fix a point p x and take a global contact form so that is asymptotically flat near p then x t 1 0 x is a pseudohermitian 3 manifold let g p c x p g p 0 with g p x x p 2 near p where x y denotes the natural pseudohermitian distance on x consider the new pseudohermitian 3 manifold with a blow up of contact form x p t 1 0 x g p 2 and let b denote the corresponding kohn laplacian on x p in this paper we prove that the weighted kohn laplacian g p 2 b has closed range in l 2 with respect to the weighted volume form g p 2 d and that the associated partial inverse and the szego projection enjoy some regularity properties near p as an application we prove the existence of some special functions in the kernel of b that grow at a specific rate at p the existence of such functions provides an important ingredient for the proof of a positive mass theorem in 3 dimensional cr geometry by cheng malchiodi and yang 5

a positive mass theorem in three dimensional cauchy riemann geometry in this note we summarize the results from 6 on the positive mass problem in 3 dimensional cr cauchy riemann geometry

pseudo einstein structures on cr manifolds etant donne une variete cr compacte non degeneree on cherche une structure pseudohermitienne associee pour laquelle le tenseur de ricci pseudohermitien est un multiple scalaire de la forme de levi

on the geometric interpretation of the nonnegative rank the nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank one factors needed to reconstruct it exactly the problem of determining this rank and computing the corresponding nonnegative factors is difficult however it has many potential applications e g in data mining graph theory and computational geometry in particular it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program in this paper we introduce and study a related quantity called the restricted nonnegative rank we show that computing this quantity is equivalent to a problem in polyhedral combinatorics and fully characterize its computational complexity this in turn sheds new light on the nonnegative rank problem and in particular allows us to provide new improved lower bounds based on its geometric interpretation we apply these results to slack matrices and linear euclidean distance matrices and obtain counter examples to two conjectures of beasly and laffey namely we show that the nonnegative rank of linear euclidean distance matrices is not necessarily equal to their dimension and that the rank of a matrix is not always greater than the nonnegative rank of its square

on the cr analogue of obata s theorem in a pseudohermitian 3 manifold in this paper we first prove the cr analogue of obata s theorem on a closed pseudohermitian 3 manifold with zero pseudohermitian torsion secondly instead of zero torsion we have the cr analogue of li yau s eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub laplacian in a closed pseudohermitian 3 manifold with nonnegative cr paneitz operator p0 finally we have a criterion for the positivity of first eigenvalue of the sub laplacian on a complete noncompact pseudohermitian 3 manifold with nonnegative cr paneitz operator the key step is a discovery of integral cr analogue of bochner formula which involving the cr paneitz operator

a matlab based rapid method for computing lattice subspaces and vector sublattices of r n applications in portfolio insurance this paper provides the construction of a powerful and efficient computational method that translates polyrakis algorithm i a polyrakis minimal lattice subspaces trans am math soc 351 1999 4183 4203 theorem 3 19 for the calculation of lattice subspaces and vector sublattices in r n in the theory of finance lattice subspaces have been extensively used in order to provide a characterization of market structures in which the cost minimizing portfolio is price independent specifically we apply our computational method in order to solve a cost minimization problem that ensures the minimum cost insured portfolio

smooth solutions of degenerate laplacians on strictly pseudoconvex domains on considere la caracterisation des solutions lisses globales de u 0 sur un domaine general strictement pseudoconvexe ou est le laplacien pour la metrique avec forme de kakler i 2 log 1

the nonnegative rank of a matrix hard problems easy solutions using elementary linear algebra we develop a technique that leads to solutions of two widely known problems on nonnegative matrices first we give a short proof of the result by vavasis stating that the nonnegative rank of a matrix is np hard to compute this proof is essentially contained in the paper by jiang and ravikumar who discussed this topic in different terms fifteen years before the work of vavasis second we present a solution of the cohen rothblum problem on rational nonnegative factorizations which was posed in 1993 and remained open until now

computational methods in lattice subspaces of c a b with applications in portfolio insurance in this article we develop a computational method for an algorithmic process first posed by polyrakis in 1996 in order to check whether a finite collection of linearly independent positive functions in c a b forms a lattice subspace lattice subspaces are closely related to a cost minimization problem in the theory of finance that ensures the minimum cost insured portfolio and this connection is further investigated here finally we propose a computational method in order to solve the minimization problem and to calculate the minimum cost insured portfolio all of the numerical work is performed using the matlab high level language

a reverse hadamard inequality abstract let u be an n n unitary matrix with determinant equal to 1 let a be an n n real matrix with rank a 2 and entries satisfying aij 1 for 1 i j n then it follows that det a u 1 this reversal of the hadamard inequality can be obtained easily from an old result of fiedler in this article we present a different proof of this fact and discuss its ramifications

on a conformally invariant variational problem on differential forms abstract we consider a conformally invariant constrained variational problem on p differential forms for p dim m 2 2 coming from conformal geometry and the associated yamabe type system of partial differential equations it is shown that the infimum p m of the functional is not bigger than p r n and if p m p r n then p m is achieved by a c 2 p form for some 0

a note on the scaled total least squares problem in this note we present two results on the scaled total least squares problem first we discuss the relation between the scaled total least squares and the least squares problems we derive an upper bound for the difference between the scaled total least squares solution and the least squares solution and establish a quantitative relation between the scaled total least squares residual and the least squares residual second we give a perturbation analysis of the scaled total least squares problem numerical experiments in comparing our results with existing results are demonstrated

bounds for the least squares distance using scaled total least squares the standard approaches to solving overdetermined linear systems bx approx c construct minimal corrections to the data to make the corrected system compatible in ordinary least squares ls the correction is restricted to the right hand side c while in scaled total least squares stls 14 12 corrections to both c and b are allowed and their relative sizes are determined by a real positive parameter gamma as gamma rightarrow 0 the stls solution approaches the ls solution our paper 12 analyzed fundamentals of the stls problem this paper presents a theoretical analysis of the relationship between the sizes of the ls and stls corrections called the ls and stls distances in terms of gamma we give new upper and lower bounds on the ls distance in terms of the stls distance compare these to existing bounds and examine the tightness of the new bounds this work can be applied to the analysis of iterative methods which minimize the residual norm and the generalized minimum residual method gmres 15 is used here to illustrate our theory

fundamental solutions for non divergence form operators on stratified groups we construct the fundamental solutions and for the non divergence form operators i j a i j x t x i x j t and i j a i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we also provide gaussian estimates of r and its derivatives and some results for the relevant cauchy problem suitable long time estimates of allow us to construct using both t saturation and approximation arguments

the super order dual of an ordered vector space and the riesz kantorovich formula a classical theorem of f riesz and l v kantorovich asserts that if l is a vector lattice and f and g are order bounded linear functionals on l then their supremum least upper bound f v g exists in l and for each x l it satisfies the so called riesz kantorovich formula f g x sup f y g z y z l and y z x related to the riesz kantorovich formula is the following long standing problem if the supremum of two order bounded linear functionals f and g on an ordered vector space exists does it then satisfy the riesz kantorovich formula in this paper we introduce an extension of the order dual of an ordered vector space and provide some answers to this long standing problem the ideas regarding the riesz kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium the techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the abovementioned problem and to the properties of the riesz kantorovich formula

lattice subspaces of c 0 1 and positive bases abstract a closed vector subspace of a vector lattice e is a lattice subspace of e if x with the induced ordering is a vector lattice in its own right but not necessarily a vector sublattice in this work we remark that c 0 1 is a universal banach lattice in the sense that every separable banach lattice is order isomorphic to a closed lattice subspace of c 0 1 in addition we show that up to an order isomorphism if a closed lattice subspace of c 0 1 has a positive basis bn then for each n there exists a subinterval jn on which bn is positive and every other element of this basis vanishes by means of our main result we present necessary and sufficient conditions that guarantee the existence of positive bases in an arbitrary lattice subspace of c 0 1 these results are related to the general existence problem of positive bases in banach lattices as well as the existence of unconditional bases in banach spaces

cr yamabe constant cr yamabe flow and its soliton abstract on a compact strictly pseudoconvex cr manifold m we consider the cr yamabe constant of its infinite conformal covering by using the maximum principles we then prove a uniqueness theorem for the cr yamabe flow on a complete noncompact cr manifold finally we obtain some properties of the cr yamabe soliton on complete noncompact cr manifolds

svd based initialization a head start for nonnegative matrix factorization we describe nonnegative double singular value decomposition nndsvd a new method designed to enhance the initialization stage of nonnegative matrix factorization nmf nndsvd can readily be combined with existing nmf algorithms the basic algorithm contains no randomization and is based on two svd processes one approximating the data matrix the other approximating positive sections of the resulting partial svd factors utilizing an algebraic property of unit rank matrices simple practical variants for nmf with dense factors are described nndsvd is also well suited to initialize nmf algorithms with sparse factors many numerical examples suggest that nndsvd leads to rapid reduction of the approximation error of many nmf algorithms

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

scaled total least squares fundamentals the standard approaches to solving overdetermined linear systems bx approx c construct minimal corrections to the vector c and or the matrix b such that the corrected system is compatible in ordinary least squares ls the correction is restricted to c while in data least squares dls it is restricted to b in scaled total least squares stls 22 corrections to both c and b are allowed and their relative sizes depend on a real positive parameter gamma stls unifies several formulations since it becomes total least squares tls when gamma 1 and in the limit corresponds to ls when gamma rightarrow 0 and dls when gamma rightarrow infty this paper analyzes a particularly useful formulation of the stls problem the analysis is based on a new assumption that guarantees existence and uniqueness of meaningful stls solutions for all parameters gamma 0 it makes the whole stls theory consistent our theory reveals the necessary and sufficient condition for preserving the smallest singular value of a matrix while appending or deleting a column this condition represents a basic matrix theory result for updating the singular value decomposition as well as the rank one modification of the hermitian eigenproblem the paper allows complex data and the equivalences in the limit of stls with dls and ls are proven for such data it is shown how any linear system bx approx c can be reduced to a minimally dimensioned core system satisfying our assumption consequently our theory and algorithms can be applied to fully general systems the basics of practical algorithms for both the stls and dls problems are indicated for either dense or large sparse systems our assumption and its consequences are compared with earlier approaches

the cr paneitz operator and the stability of cr pluriharmonic functions abstract we give a condition which ensures that the paneitz operator of an embedded three dimensional cr manifold is nonnegative and has kernel consisting only of the cr pluriharmonic functions our condition requires uniform positivity of the webster scalar curvature and the stability of the cr pluriharmonic functions for a real analytic deformation as an application we show that the real ellipsoids in c 2 are such that the cr paneitz operator is nonnegative with kernel consisting only of the cr pluriharmonic functions

renormalized chern gauss bonnet formula for complete kahler einstein metrics we present a renormalized gauss bonnet formula for a strictly pseudoconvex manifold with a complete k a hler metric given by a globally defined potential function near the boundary when the metric is asymptotically einstein the boundary contribution in the formula is explicitly written down in terms of the pseudo hermitian geometry of the boundary and is shown to be a global cr invariant the cr invariant generalizes the burns epstein invariant on 3 dimensional cr manifolds

nonnegative ranks decompositions and factorizations of nonnegative matrices the nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank one matrices into which the matrix can be decomposed additively such decompositions are useful in diverse scientific disciplines we obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm

convergence of the cr yamabe flow we consider the cr yamabe flow on a compact strictly pseudoconvex cr manifold m of real dimension 2n 1 we prove convergence of the cr yamabe flow when n 1 or m is spherical

existence of extremal functions for the stein weiss inequalities on the heisenberg group abstract in this paper we establish the existence of extremals for two kinds of stein weiss inequalities on the heisenberg group more precisely we prove the existence of extremals for the stein weiss inequalities with full weights in theorem 1 1 and the stein weiss inequalities with horizontal weights in theorem 1 4 different from the proof of the analogous inequality in euclidean spaces given by lieb 26 using riesz rearrangement inequality which is not available on the heisenberg group we employ the concentration compactness principle to obtain the existence of the maximizers on the heisenberg group our result is also new even in the euclidean case because we don t assume that the exponents of the double weights in the stein weiss inequality 1 1 are both nonnegative see theorem 1 3 and more generally theorem 1 5 therefore we extend lieb s celebrated result of the existence of extremal functions of the stein weiss inequality in the euclidean space to the case where the exponents are not necessarily both nonnegative see theorem 1 3 furthermore since the absence of translation invariance of the stein weiss inequalities additional difficulty presents and one cannot simply follow the same line of lions idea to obtain our desired result our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities theorem 1 5

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

normal cr structures on compact 3 manifolds we study normal cr compact manifolds in dimension 3 for a choice of a cr reeb vector field we associate a sasakian metric on them and we classify those metrics as a consequence the underlying manifolds are topologically finite quotients of s 3 or of a non flat circle bundle over a riemann surface of positive genus in the latter case we prove that their cr automorphisms group is a finite extension of s 1 and we classify the normal cr structures on these manifolds

computing symmetric nonnegative rank factorizations an algorithm is described for the nonnegative rank factorization nrf of some completely positive cp matrices whose rank is equal to their cp rank the algorithm can compute the symmetric nrf of any nonnegative symmetric rank r matrix that contains a diagonal principal submatrix of that rank and size with leading cost o rm2 operations in the dense case the algorithm is based on geometric considerations and is easy to implement the implications for matrix graphs are also discussed

computational methods in portfolio insurance abstract in this article we develop a computational method for an algorithmic process first posed by abramovich aliprantis polyrakis in 1994 in order to check whether a finite collection of linearly independent positive vectors in r m forms a lattice subspace lattice subspaces are closely related to a cost minimization problem in the theory of finance that ensures the minimum cost insured portfolio and this connection is further investigated here finally we propose a computational method in order to solve the minimization problem and to calculate the minimum cost insured portfolio all of the numerical work is performed using the matlab high level language

harnack inequality for non divergence form operators on stratified groups we prove lower bounds for the fundamental solutions of the non divergence form operators j i j x t x i x j t and i j i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we then prove an invariant harnack inequality for such operators as a byproduct we also study some relevant properties of the green functions on bounded domains

computation of vector sublattices and minimal lattice subspaces of rk applications in finance abstract in this article we carry out a computational study on polyrakis algorithms i a polyrakis finite dimensional lattice subspaces of c and curves of r n trans am math soc 348 1996 2793 2810 i a polyrakis minimal lattice subspaces trans am math soc 351 1999 4183 4203 these algorithms can be utilised for the determination of the vector sublattice and the minimal lattice subspace generated by a finite set of positive vectors of r k it is demonstrated that the findings of our work can be very useful in the field of economics especially in completion by options of security markets and portfolio insurance

the fefferman metric and pseudo hermitian invariants c fefferman has shown that a real strictly pseudoconvex hypersurface in complex n space carries a natural conformai lorentz metric on a circle bundle over the manifold this paper presents two intrinsic constructions of the metric valid on an abstract cr manifold one is in terms of tautologous differential forms on a natural circle bundle the other is in terms of webster s pseudohermitian invariants these results are applied to compute the connection and curvature forms of the fefferman metric explicitly

the dirichlet problem for the kohn laplacian on the heisenberg group ii for x y t rn rn r denote xj xj 2yj t yj yj 2xj t and l 14 j 1nxj2 yj2 t when n 2q la represents the action of the kohn laplacian b on q forms on the heisenberg group for n n we construct a parametrix for the dirichlet problem in smooth domains d near non characteristic points of d a point w of d is non characteristic if one of x1 xn y1 yn is transverse to d at w this yields sharp local estimates in the dirichlet problem in the appropriate non isotropic lipschitz classes the main new tool is a convolution calculus of pseudo differential operators that can be applied to the relevant layer potentials for which the usual asymptotic composition formula is false characteristic points are treated in part ii

the nmf problem and lattice subspaces abstract suppose that a is a nonnegative n m real matrix the nmf problem is the determination of two nonnegative real matrices f v so that a f v with intermediate dimension p smaller than m i n n m in this article we present a general method for the determination of two nonnegative real factors f v of a based on the theory of lattice subspaces and positive bases

an integral type brezis nirenberg problem on the heisenberg group abstract this paper is devoted to a class of integral type brezis nirenbreg problems on the heisenberg group they are a class of nonlinear integral equations on the bounded domains of heisenberg group and related to the cr yamabe problems on cr manifold based on the sharp hardy littlewood sobolev inequalities on the heisenberg group the nonexistence and existence results are obtained

uniformization of spherical cr manifolds abstract let m be a closed compact with no boundary spherical cr manifold of dimension 2 n 1 let m be the universal covering of m let denote a cr developing map m s 2 n 1 where s 2 n 1 is the standard unit sphere in complex n 1 space c n 1 suppose that the cr yamabe invariant of m is positive then we show that is injective for n 3 in the case n 2 we also show that is injective under the condition s m 1 where s m means the minimum exponent of the integrability of the green s function for the cr invariant sublaplacian on m it then follows that m is uniformizable

algebraic relations between the total least squares and least squares problems with more than one solution this paper completes our previous discussion on the total least squares tls and the least squares ls problems for the linear systemax b which may contain more than one solution 12 13 generalizes the work of golub and van loan 1 2 van huffel 8 van huffel and vandewalle 11 the tls problem is extended to the more general case the sets of the solutions and the squared residuals for the tls and ls problems are compared the concept of the weighted squares residuals is extended and the difference between the tls and the ls approaches is derived the connection between the approximate subspaces and the perturbation theories are studied it is proved that under moderate conditions all the corresponding quantities for the solution sets of the tls and the modified ls problems are close to each other while the quantities for the solution set of the ls problem are close to the corresponding ones of a subset of that of the tls problem

convergence of the yamabe flow for arbitrary initial energy we consider the yamabe flow frac partial g partial t r g r g g where g is a riemannian metric on a compact manifold m r g denotes its scalar curvature and r g denotes the mean value of the scalar curvature we prove convergence of the yamabe flow if the dimension n satisfies 3 leq n leq 5 or the initial metric is locally conformally flat

embeddability for three dimensional cr manifolds a question of principal interest in the theory of compact three dimensional cr manifolds is to understand when a given strictly pseudoconvex cr structure can be realized by an embedding in c we call a structure for which this is possible embeddable and nonembeddable otherwise this question is only of interest in three dimensions as a theorem of boutet de monvel states that any strictly pseudoconvex cr structure on a compact 2d 1 manifold is realizable as an embedding in some c provided d 1 on the other hand there is a classical example of rossi which shows that an arbitrarily small real analytic perturbation of the standard structure on the three sphere may fail to be embeddable ros later nirenberg constructed w perturbations for which the associated ob equation fails to have even local solutions n in fact this situation is generic in the f topology it is now understood principally from the work of eliashberg that cr structures with

a global invariant for three dimensional cr manifolds in this note we define a global r valued invariant of a compact strictly pseudoconvex 3 dimensional cr manifold m whose holomorphic tangent bundle is trivial the invariant arises as the evaluation of a derham cohomology class on the fundamental class of the manifold to construct the relevant form we start with the cr structure bundle y over m see ch mo whose notation we follow the form is a secondary characteristic form of this structure by fixing a contact form and coframe i e a section of y we obtain a form on m surprisingly this form is well defined up to an exact term and thus its cohomology class is well defined in h 3 m r our motivation for studying this invariant was its analogy with the r z secondary characteristic number associated by chern and simons to the conforreal class of a riemannian 3 manifold n which provides an obstruction to the conformal immersion of n in r 4 though several formal analogies to the conformal case are valid for our invariant this one does not hold up specifically in w below we calculate examples which show that the cr invariant can take on any positive real value for hypersurfaces embedded in c 2 it is also clear that the invariant is neither a homotopy nor concordance invariant but depends in an elusive way on the cr structure our inspiration came from the seminal papers of chern and moser and chern and simons the idea of looking at secondary characteristic forms of higher order geometric structures in general appears in ko oc though with a different intention in w 2 we will quickly review the definition of a cr structure the construction of y and its reduction to a pseudo hermitian structure ld webster we in w 3 we define the invariant and prove that it is in fact r valued and not r z valued as in the riemannian case we also prove that if the invariant is stationary as a function of the cr structure then m is locally cr equivalent to the standard three sphere in c 2 paralleling a result of chern and simons as noted already w 4 is devoted to the calculation of several examples

geometric inequalities on locally conformally flat manifolds through the study of some elliptic and parabolic fully nonlinear pdes we establish conformal versions of the quermassintegral inequality the sobolev inequality and the moser trudinger inequality for the geometric quantities associated to the schouten tensor on locally conformally flat manifolds

wormholes in ach einstein manifolds we give a new construction of einstein manifolds which are asymptotically complex hyperbolic inspired by the work of mazzeo pacard in the real hyperbolic case the idea is to develop a gluing theorem for 1 handle surgery at infinity which generalizes the klein construction for the complex hyperbolic metric