minors of the moore penrose inverse abstract let q k n 1 k 1 1 k n denote the strictly increasing sequences of k elements from 1 n for q k n we denote by a the submatrix of a with rows indexed by columns by the submatrix obtained by deleting the rows and columns is denoted by a lsqb rsqb for nonsingular a r n n the jacobi identity relates the minors of the inverse a 1 to those of a det a 1 lsqb rsqb 1 k i 1 i k i 1 i det a lsqb rsqb det a for any q k n we generalize the jacobi identity to matrices a r m n r expressing the minors of the moore penrose inverse a in terms of the minors of the maximal nonsingular submatrices a ij of a in our notation det a lsqb rsqb 1 vol 2 a i j n det a ij a a ij for any q k m q k n 1 k r where vol 2 a denotes the sum of squares of determinants of r r submatrices of a we apply our results to questions concerning the nonnegativity of principal minors of the moore penrose inverse

minimal entropy and collapsing with curvature bounded from below we show that if a closed manifold m admits an structure not necessarily polarized possibly of rank zero then its minimal entropy vanishes in particular this is the case if m admits a non trivial s1 action as a corollary we obtain that the simplicial volume of a manifold admitting an structure is zero we also show that if m admits an structure then it collapses with curvature bounded from below this in turn implies that m collapses with bounded scalar curvature or equivalently its yamabe invariant is non negative we show that structures of rank zero appear rather frequently every compact complex elliptic surface admits one as well as any simply connected closed 5 manifold we use these results to study the minimal entropy problem we show the following two theorems suppose that m is a closed manifold obtained by taking connected sums of copies of s4 p2 overline mathbb c p 2 s2 s2and the k3 surface then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s4 p2 s2 s2 p2 overline mathbb c p 2 or p2 p2 finally suppose that m is a closed simply connected 5 manifold then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s5 s3 s2 then on trivial s3 bundle over s2 or the wu manifold su 3 so 3

outer inverses and jacobi type identities abstract we consider matrices over a commutative ring and characterize the class of outer inverses for which jacobi type identities can be extended we obtain a necessary and sufficient condition for the existence of rao regular drazin inverse in terms of sum of principal minors of a k for some k also we obtain determinantal formula for the rao regular drazin inverse conjectures are formulated which give expressions for outer inverse and the conjectures are proved in some special cases

outer inverses jacobi type identities and nullities of submatrices abstract according to the jacobi identity if a is an invertible matrix then any minor of a 1 equals up to a sign the determinant of a 1 times the complementary minor in the transpose of a the identity is extended to any outer inverse thereby generalizing several results in the literature for special generalized inverses a permanental analog of the jacobi identity is proved bounds are obtained for the difference between the nullity of a submatrix of a and that of the complementary submatrix in any generalized inverse or an outer inverse of a the result extends earlier work of fiedler markham and gustafson for the inverse and of robinson for the moore penrose inverse

three results in connection with inverse matrices abstract first we show that the moore penrose solution of an arbitrary system of linear equations is a convex combination of the solutions of all uniquely solvable partial systems the other two results concern the elements of inverse toeplitz band matrices namely the asymptotic behavior of a determinant appearing in a formula of d s meek and a modification of a formula of w d hoskins and p j ponzo for matrices with binomial coefficients in the limit case

innovation based on gaussian elimination to compute generalized inverse at s 2 abstract in this paper we execute elementary row and column operations on the partitioned matrix g a g g g 0 into i s 0 0 0 0 0 a t s 2 to compute generalized inverse a t s 2 of a given complex matrix a where g is a matrix such that r g t and n g s the total number of multiplications and divisions operations is t m n s 2 m n 2 4 m s 1 2 n s m s n s m n s and the upper bound of t m n s is less than 6 m n 2 3 2 n 3 1 2 n 2 when n m a numerical example is shown to illustrate that this method is correct

invariant conformal metrics on ⁿ in this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its schouten tensor also we study conformal metrics on the sphere which are invariant by a k parameter subgroup of conformal diffeomorphisms of the sphere giving a bound on its maximum dimension moreover we classify conformal metrics on the sphere whose eigenvalues of the shouten tensor are all constant we call them isoparametric conformal metrics and we use a clas

integral action in output feedback for multi input multi output nonlinear systems we address a particular problem of output regulation for multi input multi output nonlinear systems specifically we are interested in making the stability of an equilibrium point and the regulation to zero of an output robust to small unmodelled discrepancies between design model and actual system in particular those introducing an offset we propose a novel procedure which is intended to be relevant to realistic scenarios as illustrated by a non academic example

determinantal representation of weighted generalized inverses abstract in this article we introduce new determinantal representations of the weighted generalized inverse by using a lemma of robinson d w robinson the classical adjoint linear algebra appl 411 2005 254 276 and using generalized cramer s rule we get the method for computing minors of the weighted moore penrose inverse and the w weighted drazin inverse

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

on the symmetric scalar curvature problem on sn abstract we discuss some existence results dealing with the scalar curvature problem on s n in the presence of various symmetries

rotation hypersurfaces in spaces of constant curvature rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed a local characterization of such hypersurfaces with dimensions greater than two is given in terms of principal curvatures some special cases of rotation hypersurfaces with constant mean curvature in hyperbolic space are studied in particular it is shown that the well known conjugation between the helicoid and the catenoid in euclidean three space extends naturally to hyperbolic three space h3 in the latter case catenoids are of three different types and the explicit correspondence is given it is also shown that there exists a family of simply connected complete embedded nontotally geodesic stable minimal surfaces in h3

surgery in dimension four and noncompact 5 manifolds this paper describes a precise relationship between the problems of completing surgery in dimension four and finding boundaries for noncompact 5 manifolds two outstanding problems of low dimensional topology are completing 4 dimensional surgery and finding boundaries for noncompact 5 dimensional manifolds although no complete theory is known for either problem some interesting results have been found see 1 4 5 10 and addendum in this paper we describe a complete relationship between these two problems consequently knowledge of 4 dimensional surgery yields information about noncompact 5 manifolds and conversely all manifolds and maps in this paper are assumed to be smooth each manifold is provided with a basepoint and when appropriate basepaths will be chosen to given 2 spheres without explicit mention i wish to thank gerald a anderson and frank quinn for helpful discussions also i wish to thank patrick gilmer for his advice and encouragement during this project 1 statement of results let f b m am y x be a degree one normal map in the sense of wall 11 where m is a compact connected 4 manifold and y x is a poincare pair not necessarily simple if x 0 then we require that f jam am x induces an isomorphism of homology groups with a z vi y coefficients also we assume that f is 2 connected k2 m is a free a module and the surgery obstruction a f b e lh 7 c is zero if h c k2 m is a subkernel we will say that h is representable in m if we can find disjoint framed embedded 2 spheres si sr c int m such that the inclusion m ui si m induces an isomorphism of fundamental groups and the classes of sl sr in h2 m a 7t2 m form a basis over a for h the subkernel h is stably representable if for some integer k 0 h ed kei ek is representable in m k s2 x s2 here ei denotes the class of s2 x 1 c ith summand s2 x s2 and e1 ek is the submodule generated by el ek we regard h ed ke ek as received by the editors august 10 1982 1980 mathematics subject classification primary 57r65 secondary 57m99 1985 american mathematical society 0002 9947 85 1 00 25 per page

theory of ordered spaces this is the first of a planned series of investigations on the theory of ordered spaces based upon four axioms two of these the order i 1 1 and the local structure ii 5 1 axioms provide the structure of the theory and the other two the identification i 1 11 and cone i 2 7 axioms eliminate pathologies or excessive generality in the present paper the axioms are supplemented by the nontriviality conditions i 1 9 and a regularity property ii 4 2

a volume associated with m x n matrices abstract let a mxnr with nonzero singular values 1 2 r the volume of a vol a is defined as zero if r 0 and otherwise vola ri 1 i or equivalently vola det2aij summing over all r x r nonsingularsubmatrices aij the matrix volume vol a generalizes the absolute value of the determinant from nonsingular to arbitrary matrices any r dimensional unit cube in r at is mapped under a into a parallelepiped of volume vol a this paper gives properties and applications of vol a in particular the moore penrose inverse of a is a convex combination of ordinary inverses of its maximal nonsingular submatrices a i j ij or is a convex combination of moore penrose inverses of its maximal full rank submatrices a i i j j where the convex weights are proport ional to the squares of volumes of the corresponding submatrices since a a the matrix a is a convex combination of inverses of submatrices of a interestingly the weights of these convex combinations are the above weights in transposed position

the global structure of simple space times according to a standard definition of penrose a space time admitting well defined future and past null infinitiesi andi is asymptotically simple if it has no closed timelike curves and all its endless null geodesics originate fromi and terminate ati the global structure of such space times has previously been successfully investigated only in the presence of additional constraints the present paper deals with the general case it is shown thati is diffeomorphic to the complement of a point in some contractible open 3 manifold the strongly causal regioni0 ofi is diffeomorphic to mathbb s 2 times mathbb r and every compact connected spacelike 2 surface ini is contained ini0 and is a strong deformation retract of bothi0 andi moreover the space time must be globally hyperbolic with cauchy surfaces which subject to the truth of the poincare conjecture are diffeomorphic to 3

group actions corks and exotic smoothings of mathbb r 4 r 4 we provide the first information on diffeotopy groups of exotic smoothings of mathbb r 4 for each of uncountably many smoothings there are uncountably many isotopy classes of self diffeomorphisms we realize these by various explicit group actions there are also actions at infinity by nonfinitely generated groups for which no nontrivial element extends over the whole manifold in contrast every diffeomorphism of the end of the universal mathbb r 4 extends our techniques apply to many other open 4 manifolds and are related to cork theory we show that under broad hypotheses cork twisting is equivalent up to blowups to twisting on an exotic mathbb r 4 and give applications

an equation of monge ampere type in conformal geometry and four manifolds of positive ricci curvature we formulate natural conformally invariant conditions on a 4 manifold for the existence of a metric whose schouten tensor satisfies a quadratic inequality this inequality implies that the eigenvalues of the ricci tensor are positively pinched

full rank representation of generalized inverse at s 2 and its application in this paper we introduce a full rank representation of the generalized inverse a t s 2 of a given complex matrix a which is based on an arbitrary full rank decomposition of g where g is a matrix such that r g t and n g s using this representation we introduce the minor of the generalized inverse a t s 2 as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained as an application we use an example to demonstrate that this representation is correct

a family of freely slice good boundary links we show that every good boundary link with a pair of derivative links on a seifert surface satisfying a homotopically trivial plus assumption is freely slice this subsumes all previously known methods for freely slicing good boundary links with two or more components and provides new freely slice links

scalar curvature functions in a conformal class of metrics and conformal transformations on etudie le probleme de prescrire la courbure scalaire dans une classe conforme pour l action du groupe conforme on montre l universalite des conditions d integrabilite dues a j l kazdan et f w warner

generalized inverses with nonnegative principal minors abstract conditions for a g inverse to be a p0 matrix are obtained in terms of bordered matrices some results are derived as corollary in the case of matrices of order n and rank n 1 a sufficient condition is given for the moore penrose inverse of i t to be a p0 matrix where t is the transition probability matrix of a markov chain

generalized inverses over integral domains in an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r r minors of the matrix is one in the same paper a procedure for constructing a generalized inverse from such a linear combination was also given in the present paper we show that any reflexive generalized inverse can be obtained by that procedure we also show that if the integral domain is a principal ideal domain every generalized inverse can be obtained by that procedure it is also shown that a matrix a of rank r over an integral domain has moore penrose inverse if and only if the sum of squares of all r r minors of a is an invertible element of the integral domain

cauchy binet for pseudo determinants abstract the pseudo determinant det a of a square matrix a is defined as the product of the nonzero eigenvalues of a it is a basis independent number which is up to a sign the first nonzero entry of the characteristic polynomial of a we prove det f t g p det f p det g p for any two n m matrices f g the sum to the right runs over all k k minors of a where k is determined by f and g if f g is the incidence matrix of a graph this directly implies the kirchhoff tree theorem as l f t g is then the laplacian and det 2 f p 0 1 is equal to 1 if p is a rooted spanning tree a consequence is the following pythagorean theorem for any self adjoint matrix a of rank k one has det 2 a p det 2 a p where det a p runs over k k minors of a more generally we prove the polynomial identity det 1 x f t g p x p det f p det g p for classical determinants det which holds for any two n m matrices f g and where the sum on the right is taken over all minors p understanding the sum to be 1 if p 0 it implies the pythagorean identity det 1 f t f p det 2 f p which holds for any n m matrix f and sums again over all minors f p if applied to the incidence matrix f of a finite simple graph it produces the chebotarev shamis forest theorem telling that det 1 l is the number of rooted spanning forests in the graph with laplacian l

exotic smoothness and physics the essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space time models especially the simplest one r4 possess a rich multiplicity of such structures no two of which are diffeomorphic to each other and thus to the standard one this means that physics has available to it a new panoply of structures for space time models these can be thought of as a source of new global but not properly topological features this article reviews some background differential topology together with a discussion of the role which a differentiable structure necessarily plays in the statement of any physical theory recalling that diffeomorphisms are at the heart of the principle of general relativity some of the history of the discovery of exotic i e nonstandard differentiable structures is reviewed some new results suggesting the spatial localization of such exotic structures are described and speculations are made on the possible opportunities that such structures present for the further development of physical theories

nonnegativity of principal minors of generalized inverses of p0 matrices abstract the result of principal interest established in this paper is that if a is an n n singular irreducible m matrix then a large class of generalized inverses of a possesses the property that each of its elements has all its principal minors nonnegative the class contains both the group and the moore penrose generalized inverses of a in an application of our results it is shown that the fundamental matrix of a continuous in time ergodic markov chain on a finite state space has all its principal minors nonnegative

construction of group actions on four manifolds it is shown that any cyclic group of odd prime order acts on any closed simply connected topological 4 manifold inducing the identity on integral homology the action is locally linear except perhaps at one isolated fixed point in the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear introduction in this paper we shall show that every closed simply connected topological 4 manifold m4 admits an action of any cyclic group zp of odd prime order p the action will be homologically trivial that is induce the identity on integral homology be pseudofree that is have only isolated fixed points except in certain cases when p 3 and be locally linear except perhaps at one isolated fixed point when p 2 and the intersection form of m4 has even type then the same conclusion holds this is also due to s kwasik and p vogel 1985 by a somewhat different proof if p 2 and the intersection form of m has odd type however then an action of z2 which is homologically trivial must have a fixed point set containing two dimensional components see 7 further it is harder to control the kirby siebenmann triangulation obstruction which is then not determined by the intersection form it is interesting to ask whether these actions can be locally linear or smooth when m4 is smooth the existence of a locally linear z2 action implies the vanishing of the kirby siebenmann obstruction see kwasik and vogel 1984 we shall show however that for p 3 in general these actions can be constructed to be locally linear in a recent preprint kwasik has shown that this is the case for the fake cp2 when p is odd by a rather different proof these actions are homologically trivial and have only isolated fixed points except sometimes when p 3 the g signature theorem shows that not every simply connected 4 manifold admits pseudofree locally linear actions when p 3 we shall see for example that neither the e8 manifold the kummer surface nor a nontrivial connected sum of copies of cp2 admits such an action for p 3 in broad outline our construction goes as follows let a closed oriented simply connected 4 manifold m be given by studying equivariant framed links we construct a compact smooth 4 manifold with boundary having the intersection form of m and admitting a smooth zp action which is pseudofree and homologically trivial the action is free on the boundary homology sphere e in a paper primarily focused on equivariant plumbing diagrams for high dimensional highly connected manifolds with even intersection forms weintraub 1975 carried out most of this we give an independent development because we need to be able to better control received by the editors october 16 1985 and in revised form january 7 1986 1980 mathematics subject classification 1985 revision primary 57s17 57s25 57n15 research supported in part by a grant from the national science foundation 1987 american mathematical society 0002 9947 87 1 00 25 per page

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

a surgery sequence in dimension four the relations with knot concordance we present a systematic treatment of the classification problem for compact smooth 4 manifolds m it is modeled on the surgery exact sequence the central theorem in the classification of n manifolds n 5 the price for the extension to dimension 4 is a hole in m where a homotopy 1 skeleton should be there is no homotopy theoretic or surgical obstruction to complet ing m with a wedge of circles so that the completion has the topology of a compact smooth manifold this point set problem is all that stands between 4 manifolds and the tranquility that prevails in higher dimensions when m is simply connected only a point is missing from the model the applications of this are discussed in f and fq 1 the general theory has application to knot and link theory in particular knots with alexander polynomial 1 are characterized geometrically as knots admitting a certain

linking and causality in globally hyperbolic space times the classical linking number lk is defined when link components are zero homologous in 15 we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes here we apply alk to the study of causality in lorentzian manifolds

outer inverses of matrices abstract let r be a commutative ring with 1 and let g r n m if a r m n is of rank r 1 with tr c r ag 1 then aga a iff gcr a a that is a is an outer inverse of g iff a is the adjoint image of cr a under g

classification of singular radial solutions to the σk yamabe equation on annular domains the study of the kth elementary symmetric function of the weyl schouten curvature tensor of a riemannian metric the so called k curvature has produced many fruitful results in conformal geometry in recent years in these studies in conformal geometry the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic pde important advances have been made in recent years in the understanding of the analytic behavior of solutions of the pde however the singular behavior of these solutions which is important in describing many important questions in conformal geometry is little understood this note classifies all possible radial solutions in particular the singular solutions of the k yamabe equation which describes conformal metrics whose k curvature equals a constant although the analysis involved is of elementary nature these results should provide useful guidance in studying the behavior of singular solutions in the general situation

new proofs of two representations and minor of generalized inverse a 2 t s abstract in this paper we prove the limit representation for the generalized inverse a t s 2 of a given complex matrix a in a simpler way then using this representation two known results the generalized inverse a t s 2 as an analogs of convex combination with the minor of adja and the representation of the minor of the generalized inverse a t s 2 were derived anew as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2