some schur complement identities and applications to matrix extrapolation methods the schur complement has many applications in linear algebra and in numerical analysis schur complement identities have been used for implementing some scalar and vector extrapolation methods in this paper we define some matrix sequence transformations and give two new schur complement identities that will be used to derive algorithms for implementing these transformations

spectral shorted matrices abstract given a n n positive semidefinite matrix a and a subspace s of c n s a denotes the shorted matrix of a to s we consider the notion of spectral shorted matrix s a lim m s a m 1 m wecompletely characterize this martix in terms of s and the spectrum and the eigenspaces of a we show the relation of this notion with the spectral order of matrices and the kolmogorov s complexity of a to a vector c n

inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

on inverse positivity of sub direct sums of matrices abstract in this note the authors consider the problem of inverse positivity of k subdirect sum of matrices the main results provide a solution to an open problem posed recently

global existence for systems of parabolic conservation laws in several space variables abstract we prove the global existence of solutions of the cauchy problem for certain systems of conservation laws with artificial viscosity terms added the system is assumed to admit a quadratic entropy which is consistent with the viscosity matrix and the initial data is assumed to be close to a constant in l2 l in particular our result applies to the equations of compressible fluid flow in two and three space variables

geodesic and re coupling induced limits to s2n group invariants of uniform k1 k2n dual tensorial sets in spin physics or nmr sr dynamical structure on hv via polyhedral re coupling and time reversal invariance for s n based uniform auxiliary dual tensorial udt sets explicitly with even maximal all alike j i k i over v j 1 j 2 n or k 1 k 2 n sub rank auxiliary labelling the specific time reversal invariance dependent dual group invariants and their independent cardinalities si 2 n are fundamental to understanding simply reducible sr dynamical spin system structure over liouvillian carrier space a quantum physics property implicit in dual quasi particle qp formalisms associated with for example uniform identical multiple spin nmr systems such udt sets stand in total contrast to the distinct j i k i of j 1 j n k 1 k n sub rank orthogonal tensorial sets with their z aa graph invariants which for reasons discussed in the text are of restricted validity for udts as operator bases related to the spin interaction network based tensors of high s n permutational nmr hamiltonians liouvillians the use of qp boson superboson formalisms and their dual projective maps f p temme 1993 physica a 198 245 261 prove invaluable once the independent cardinality of the multiple invariants of these udts alias the su 2 s 2 n group invariants here denoted si 2 n are known in terms of democratic re coupling dr and polyhedral combinatorial pc sampling applied to the time reversal invariance tri properties of the uniform multiple spin sub system so the sr dual carrier spaces v follow directly as well defined entities for udts of general 2 n 6 fold ensembles the augmented models of tri discussed here i e beyond linear re coupled weyl forms are essential due to the presence of non abelian degeneracy a regular geometric automorphic limit is established for sequential 2 n based si 2 n total series specific to higher uniform j 1 j i k 1 k i based dual tensors by treating the si 2 n obtained from the dr based tris as group measures and invoking an automorphic geodesic augmentation certain otherwise inaccessible additional si 2 n and hence udts may be established these are based on sporadic higher regular 2 n 12 fold ensemble invariants and are obtained via finite group duality properties implicit in the concept of group measures the pc dr based views of general 2 n interacting spins of udts expressed here give specific insight into dynamic structure over explicit tri invariant defined liouvillian carrier spaces this lattice point erdosian view of dr shows that udt invariants are distinct and more contracted than predicted by the earlier linear re coupling based d 0 u su 2 approach the presence of degeneracy in general high 2 n udts as non abelian entities ensures that the homomorphic based d 0 u enumeration with its implied linear graphical re coupling is invalid for high 2 n uniform nmr multi spin systems whose tri properties are governed by dr re coupling the prime focus of the report concerns the positive conceptual value of pc lattice point based dr as applied to the s 2 n invariants which define udts of nmr and their completeness brief contrasts with the earlier more restricted o 2 k 1 overall invariants of o n g based democratic re coupled quantum systems are given for completeness

asymptotics of solutions of nonlinear parabolic equations abstract this paper is concerned with the large time behaviour of solutions to the cauchy problem of the following nonlinear parabolic equations u t u f u d x u d x 2 u u r n u t x t 0 u 0 x x r n n 1 under the optimal growth conditions on the smooth nonlinear function f u dxu dx2u we obtain the global existence results to the above cauchy problem the influence of the nonlinear function f u dxu dx2u on the large time behaviour of the global smooth solution u t x is also studied

remarks on the schur complement abstract first a generalization of the schur determinantal formula is given using properties of quasidirect sums of matrices a new characterization of the schur complement is proved

on complementable operators in the sense of t ando abstract consider an operator a h k between hilbert spaces and closed subspaces s h and t k if there exist projections e on h and f on k such that r e s r f t and a e f a then a is called s t complementable the origin of this notion comes from the idea of t ando of defining schur complements in terms of operators in this paper we present some characterizations of these triples a s t and applications to bilateral schur complements and generalized wiener hopf operators

global existence and asymptotics of solutions of the cahn hilliard equation abstract this paper is concerned with the cauchy problem of the cahn hilliard equation u t u 2 u 0 x r n t 0 u t 0 u 0 x x r n first we construct a local smooth solution u t x to the above cauchy problem then by combining some a priori estimates sobolev s embedding theorem and the continuity argument the local smooth solution u t x is extended step by step to all t 0 provided that the smooth nonlinear function u satisfies a certain local growth condition at some fixed point u r and that u 0 x u l 1 r n is suitably small secondly we show that the global smooth solution u t x satisfies the following temporal decay estimates d k u t x u l p r n c 1 t k 4 n 4 1 1 p t 0 k 0 1 here p 1 c 0 is a constant depending on and 0 is any positive constant which can be chosen sufficiently small at last we show that under a strong assumption on the growth of the nonlinear function u at u u the asymptotics of solutions of the above cauchy problem is described by u 0 t n 4 g x t 4 here 0 r n u 0 x u d x g x r n exp 4 i x d

shorted matrices an extended concept and some applications the shorting of an operator hitherto considered by krein 11 and by anderson and trapp 3 only for positive operators is extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite some applications of the shorted matrix in mathematical statistics are discussed

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

more on maximal and minimal ranks of schur complements with applications the maximal and minimal ranks of the schur complement d cgb are determined with respect to generalized inverse g of a matrix a where g is taken respectively as least squares g inverse a 1 3 minimum norm g inverse a 1 4 as well as a 1 2 3 a 1 2 4 and a 1 3 4 various consequences are also presented

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

a class of convergent finite difference schemes for certain nonlinear parabolic systems on considere le systeme des equations v t u x ev xx u t p v x u xx x t rxr avec les donnees initiales v u x 0 v 0 x u 0 x x r

non euclidean geometrical aspects of the schur and levinson szego algorithms in this paper we address non euclidean geometrical aspects of the schur and levinson szego algorithms we first show that the lobachevski geometry is by construction one natural geometrical environment of these algorithms since they necessarily make use of automorphisms of the unit disk we next consider the algorithms in the particular context of their application to linear prediction then the schur resp levinson szego algorithm receives a direct resp polar spherical trigonometry st interpretation which is a new feature of the classical duality of both algorithms

complementable and almost definite matrices abstract in recent papers 1 and 5 m complementable matrices were defined by t ando and almost definite matrices by duffin and morley in this paper additional characterizations are given for these sets of matrices making use of the generalized inverse also rank properties and generalized inverses of these matrices are discussed

some inequalities on generalized schur complements abstract this paper presents some inequalities on generalized schur complements let a be an n n hermitian positive semidefinite matrix denote by a the generalized schur complement of a principal submatrix indexed by a set in a let a be the moore penrose inverse of a and a be the eigenvalue vector of a the main results of this paper are 1 a a where is the complement of in 1 2 n 2 a r r a for any real number r 1 3 c ac c a c for any matrix c of certain properties on partitioning

multiple mode diffusion waves for viscous nonstrictly hyperbolic conservation laws we study the large time behaviors of solutions of viscous conservation laws whose inviscid part is a nonstrictly hyperbolic system the initial data considered here is a perturbation of a constant state it is shown that the solutions converge to single mode diffusion waves in directions of strictly hyperbolic fields and to multiple mode diffusion waves in directions of nonstrictly hyperbolic fields the multiple mode diffusion waves which are the new elements here are the self similar solutions of the viscous conservation laws projected to the nonstrictly hyperbolic fields with the nonlinear fluxes replaced by their quadratic parts the convergence rate to these diffusion waves iso t 3 4 1 2p inl p 1 p with 0 being arbitrarily small

expression of the drazin and mp inverse of partitioned matrix and quotient identity of generalized schur complement in this paper we give representations of the drazin and mp inverse of a 2x2 block matrix and quotient identities for the generalized schur complement of a partitioned 3x3 matrix under conditions different than those used in recent papers on the subject we present numerical examples to illustrate our results

intensive structure of heisenburg supergenerator algebras over hv su2 dual carrier subspaces in liouville nmr formalisms superbosons as unit tensor superoperators abstract the structure of supergenerator algebra over simply reducible carrier space h v associated with a lie algebra over the domain of tensorial bases of nmr density operator formalisms for spin coherence are set out in terms of s 2 i s i s j s 2 j or s 2 i superbosonic forms corresponding to six unit tensors or fundamental wigner superoperators the kernel defining the characteristic null space of these unit tensors is also given some extensions of these concepts to the realm of many body nmr spin cluster problems and to rota like invariants over dual spin space are summarise the s 2 i set of component s bosons are shown to generate a group algebra and to display the requisite ladder aspects of the i set of superoperators in this respect the underlying premise yields an improved insight into the sets of su2 and su2 x s n tensors associated with nmr formalisms for spin dynamics or for model realizations of coherence transfer or relaxation by providing the intensive substructure of liouville carrier space the mapping aspects within which it has been shown that each h v is a simply reducible carrier subspace is seen as an essential preliminary aspect of su2 duality over liouville space in understanding spin cluster systems in nmr the value of p kq s 2 i monomial formalisms in appreciating the structure of liouville space and its associated transformational properties is stressed

bilateral shorted operators and parallel sums in this paper we study shorted operators relative to two different subspaces for bounded operators on infinite dimensional hilbert spaces we define two notions of complementability in the sense of ando for operators and study the properties of the shorted operators when they can be defined we use these facts in order to define and study the notions of parallel sum and substraction in this hilbertian context

an upper bound on the number of linear relations identified from noisy data by the frisch scheme abstract the frisch scheme is one of many approaches to the identification of linear relations from noisy data solution of the associated matrix optimization problem leads to the identification of a maximal number of such relations to date no general solution procedure is known here an upper bound on the maximal number of relations that can be identified is presented the upper bound is derived using the theory of schur complements in conjunction with a result for the case where only one relation can be identified the possibility of developing a completely general solution procedure using these ideas is also briefly discussed

beyond simple point sets to the indistinguishable point set tensorial limits of r w algebras multiple g invariants and the carrier map based quantal basis completeness of uniform nmr spin systems augmented quasiparticle qp mappings as applied to indistinguishable point sets of liouvillian democratic recoupled dr tensors provide for a 1 1 invariant labelling of the underlying disjoint dual projective map carrier subspaces where the liouville pattern basis set is defined via superboson unit tensor actions on a null space the co operative action liouville algebras described here imply parallel limitations to jucys graph recoupling and its related racah wigner r w algebras once dr indistinguishable point tensorial sets are involved as in non sr dominant nmr spin symmetry the importance of g invariants as labels for disjoint carrier subspaces in such automorphic spin symmetries arises from their essential role in defining the quantal completeness of indistinguishable point sets from the established properties of augmented qps as super bosons temme 2002 int j quantum chem 89 429 i e beyond the earlier hilbert space based louck and biedenharn boson pattern views insight into atiyah and sutcliffe s a s assertions atiyah and sutcliffe 2002 proc r soc a 458 1089 on the limitations of graph recoupling theory to distinct point sets is obtained this clarifies the wider analytic intractible of automorphic dr spin systems beyond the l vi civit cyclic commutation r w approach l vy leblond and l vy nahas 1965 j math phys 6 1372 which holds for a mono invariant problem for rotating frame density matrix approaches to a n a n x and dual nmr systems the focus is necessarily on the specialized nature of indistinguishable point sets within multiple invariant theoretic based dynamical spin physics here the gi s gi cardinality constitute an important part of the dual irrep set with combinatorics as a central facet of invariant theory playing a crucial role in the concept of quantal completeness and the impact of a s s assertion on the existing nmr tensorial physics clearly the role of liouvillian yamanouchi projection now as disjoint subspatial based transformational properties defines such dr bases and their unit tensors a brief outline is given of the structure of augmented general indexed l vi civit superoperators with their multiple gi labelled carrier subspaces

uniform decay rated for parabolic conservations laws on etablit des taux de decroissance uniformes optimaux pour des solutions du probleme de cauchy avec donnees grandes pour des lois de conservation paraboliques scalaires a plusieurs dimensions d espace

convergence to diffusion waves of solutions for viscous conservation laws we study the large time behavior of solutions of viscous conservation laws it is shown that solutions tend to diffusion waves which are constructed based on the heat equation and burgers equation the convergence is in thelp 1 p sense and is obtained as a consequence of thel2 decay of the difference between the solution and its asymptotic state of diffusion waves

nonlinear stability and optimal decay rate for a multidimensional generalized kuramoto sivashinsky system we study the cauchy problem for a generalized kuramoto sivashinsky system in multidimension the zero equilibrium is linearly stable for coefficients within a certain range we further establish the global existence nonlinear stability and optimal decay rate of the solution for coefficients within the same range though with suitable restriction to the initial data c 2000 academic press

a note on the optimal temporal decay estimates of solutions to the cahn hilliard equation abstract this paper is concerned with the optimal temporal decay estimates on the solutions of the cauchy problem of the cahn hilliard equation it is shown in liu wang and zhao 2007 11 that such a cauchy problem admits a unique global smooth solution u t x provided that the smooth nonlinear function u satisfies a local growth condition furthermore if u satisfies a somewhat stronger local growth condition the optimal temporal decay estimates on u t x are also obtained in liu wang and zhao 2007 11 thus a natural question is how to deduce the optimal temporal decay estimates on u t x only under the local growth condition which is sufficient to guarantee the global solvability of the corresponding cauchy problem and the main purpose of this paper is devoted to this problem our analysis is motivated by the technique developed recently in ukai yang and zhao 2006 15 with a slight modification

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

brief monotonicity of the optimal cost in the discrete time regulator problem and schur complements this paper deals with the monotonicity of the optimal rather than the more usual stabilising cost of the discrete time regulator problem the approach relies on a monotonicity result on riccati operators and on a formula for the difference of schur complements

the extreme points of a set of positive semidefinite operators abstract we determine the extreme points and exposed points of the convex set of positive semidefinite operators less than a given positive semidefinite operator a on a finite dimensional space the extreme and exposed points are the same and are obtained by shorting the operator a to all subspaces of the space on which it is defined the extreme and exposed subsets are the intervals between the shorted operators corresponding to two subspaces one contained in the other

specialised nmr spin cluster symmetry chains under su m x ln group dualities role of democratic recoupling and projective geometric mapping from su m p tuples onto γ γ v ln or onto γ v ln g for 6 n fold cage clusters the inherent nature of the dual group spin algebras for nmr spin clusters over their associated carrier spaces is examined in the context of scalar invariants sis the a n su m ln g aspects of ax n cage spin biclusters furnish these si forms and focus our attention on their part in the ln induced representational hierarchy for democratic recoupling in nmr spin dynamical formalisms the purpose of the present work is to examine the nature of the specialised symmetry chains their mappings and su2 distinct hv carrier subspaces especially for non magnetically equivalent non me supermolecular spin biclusters in the 6 n 20 60 2160 limit a hierarchy of intracluster jij interactions ensures that the ln group and its graph theoretical schemata dominate the spin algebra these heisenberg aspects arise from the bilinear terms of the zeroth hamiltonian 0 their dominance highlights certain contrasts with conventional unitary algebras and wigner unit operator properties the explicit induced symmetry recoupling labels over su2 ln carrier subspaces of cluster liouville nmr formalisms temme physica a 166 1990 676 198 1993 245 allow a stepwise subduction over the dual symmetry chain from rota scalar invariants over a field or p tuples number parts of n r r r r as su m within the chain su m v ln v ln g as demonstrated e g for 6 n 12 20 60 2160 and m 3 4 6 spin systems of bh 2 n 13ch 20 or 13cf n as as member of the set of n 60z2 supermolecular fullerenes for the dodecahedral spin symmetries of projective geometric aspects of high p m m2 tuples yield further detailed insight into the nature of group duality over spin algebras of both hilbert and liouville spaces such views go beyond the physical insight derivable from the study of casimir invariants whose proper role lies elsewhere the dual symmetry chain and mapping concepts presented here are placed in the context both of graph theory of nmr spin clusters k balasubramanian j chem phys 78 1982 6369 and of some recent views on augmented boson algebras referred to above

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

inverses of 2 2 block matrices abstract in this paper the authors give explicit inverse formulae for 2 2 block matrices with three different partitions then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as hamiltonian per hermitian and centro hermitian matrices

lossless and dissipative distributed systems this paper deals with linear shift invariant distributed systems by this we mean systems described by constant coefficient linear partial differential equations we define dissipativity with respect to a quadratic differential form i e a quadratic functional in the system variables and their partial derivatives the main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate the proof of this result involves the construction of the dissipation rate we show that this problem can be reduced to hilbert s 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions

simultaneous diagonalization of rectangular matrices abstract a matrix d is said to be diagonal if its i j th element is null whenever i and j are unequal for a set a of matrices a of the same order the paper gives necessary and sufficient conditions for nonsingular matrices s and t to exist such that sa t d is diagonal for each matrix a in the set

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

global existence and optimal temporal decay estimates for systems of parabolic conservation laws ii the multidimensional case this paper is a continuation of our previous paper it is concerned with the global existence and the optimal temporal decay estimates for the cauchy problem of the following multidimensional parabolic conservation laws

new vector sequence transformations many numerical methods produce sequences of vectors converging to the solution of a problem when the convergence is slow the sequence can be transformed into a new vector sequence which under some assumptions converges faster to the same limit the construction of a sequence transformation is based on its kernel that is the set of sequences which are transformed into a constant sequence in this paper new vector sequence transformations are built from kernels which extent those of the most general transformations known so far

algebraic linearizations of matrix polynomials abstract we show how to construct linearizations of matrix polynomials z a z d 0 c 0 a z b z a z b z when deg b z deg a z and z a z d 0 b z c 0 from linearizations of the component parts a z and b z this allows the extension to matrix polynomials of a new companion matrix construction

generalized schur complements and oblique projections let s be a closed subspace of a hilbert space h and a a bounded linear selfadjoint operator on h in this note we show that the existence of a selfadjoint projections with range s is related to some properties of shorted operators schur complements in ando s generalization of the classical concept and compressions 2001 elsevier science inc all rights reserved

approximation by matrices positive semidefinite on a subspace abstract we obtain the best approximation to a matrix by matrices positive semidefinite on a subspace as a by product we present two new characterizations of euclidean distance matrices

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 case study of global stability of strong rarefaction waves for 2 2 hyperbolic conservation laws with artificial viscosity this paper is concerned with the global stability of strong rarefaction waves for a class of 2 x 2 hyperbolic conservation laws with artificial viscosity i e the p system with artificial viscosity

complementary schur complements abstract for square matrices the relationship is discussed between the notion of schur complement and the equivalent concepts of matricial coupling and equivalence after extension special attention is paid to the hermitian and skew hermitian cases