oscillation criteria for nonlinear damped dynamic equations on time scales abstract we present new oscillation criteria for the second order nonlinear damped delay dynamic equation r t x t p t x t q t f x t 0 our results generalize and improve some known results for oscillation of second order nonlinear delay dynamic equation our results are illustrated with examples

a generalized unitary hessenberg matrix this paper describes the use of a generalized isometric arnoldi algorithm to reduce a unitary matrix via unitary similarity to a product of elementary reflectors and permutations the computation is analogous to the reduction of a unitary matrix to a unitary hessenberg matrix using the isometric arnoldi algorithm in the case in which a is a shift matrix the reduction provides a novel recurrence for the factor r in the qr factorization of a toeplitz like matrix

comparison theorems for linear dynamic equations on time scales we obtain several comparison theorems for second order linear dynamic equations on a time scale these results extend comparison theorems for the continuous case and provide some new results in the discrete case as well as other more general situations

compact riemannian manifolds with positive curvature operators m is said to have positive curvature operators if the eigenvalues of z are positive at each point p m meyer used the theory of harmonic forms to prove that a compact oriented n dimensional riemannian manifold with positive curvature operators must have the real homology of an n dimensional sphere gm proposition 2 9 using the theory of minimal two spheres we will outline a proof of the following stronger result

on two generalized inverse eigenvalue problems for hessenberg upper triangular pencils and their application to the study of gmres convergence abstract we discuss two generalized inverse eigenvalue problems the first one for a given unreduced upper hessenberg matrix h find a nonsingular upper triangular matrix t such that all the pencils h k t k have prescribed eigenvalues where h k and t k are the leading k k principal submatrices of h and t respectively the second one for a given unitary unreduced upper hessenberg matrix q find a nonsingular upper triangular matrix t such that all the pencils t k q k have prescribed eigenvalues where t k is the leading k k principal submatrix of t and q k is the conjugate transpose of the leading k k principal submatrix of q we present the necessary and sufficient conditions for the solvability of the two problems our results lead to an alternative proof for the statement that any admissible ritz value set or admissible harmonic ritz value set is possible for the prescribed gmres residual norms here the term admissible means there are some restrictions on the sets if gmres stagnates at some iterations

new kamenev type oscillation criteria for second order nonlinear advanced dynamic equations this paper presents a new kamenev type oscillation criterion for a second order advanced dynamic equation on a time scale two examples are included to show that the result obtained complements and improves those reported in the literature

oscillation criteria for a class of second order emden fowler delay dynamic equations on time scales abstract by means of riccati transformation technique we establish some new oscillation criteria for the second order emden fowler delay dynamic equations x t p t x t 0 on a time scale t here is a quotient of odd positive integers with p t real valued positive rd continuous functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales our results in this paper not only extend the results given in r p agarwal m bohner s h saker oscillation of second order delay dynamic equations can appl math q 13 1 2005 1 18 but also unify the oscillation of the second order emden fowler delay differential equation and the second order emden fowler delay difference equation

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

an extended qr solver for large profiled matrices a new method for the solution of the standard eigenvalue problem with large symmetric profile matrices is presented the method is based on the well known qr method for dense matrices a new flexible and reliable extension of the method is developed that is highly suited for the independent computation of any set of eigenvalues in order to analyze the weak convergence of the method in the presence of clustered eigenvalues the qr method is studied two effective stable and numerically cheap extensions are introduced to overcome the troublesome stagnation of the convergence a repeated preconditioning process in combination with jacobi rotations in the parts of the matrix with the strongest convergence is developed to significantly improve both local and global convergence the extensions preserve the profile structure of the matrix the efficiency of the new method is demonstrated with several examples

minimal two spheres in kahler einstein fano manifolds abstract in this paper we show the existence of stable symplectic non holomorphic two spheres in kahler manifolds of positive constant scalar curvature of real dimension four and in kahler einstein fano manifolds of real dimension six some of the techniques used involve deformation theory of algebraic cycles

sensitivity analysis for szegő polynomials szeg polynomials are orthogonal with respect to an inner product on the unit circle numerical methods for weighted least squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of szeg polynomials this paper discusses the conditioning of several mappings involving szeg polynomials and thereby sheds light on the sensitivity of some approximation problems involving trigonometric polynomials

oscillation criteria for two dimensional dynamic systems on time scales we establish some oscillation criteria for the two dimensional dynamic system x d t b t g y s t y d t a t f x t on a time scale t our results not only unify the oscillation of two dimensional differential systems and difference systems but include the oscillation results for differential systems and provide new oscillation criteria for difference systems several examples are considered to illustrate the main results

oscillation criteria for second order nonlinear delay dynamic equations on time scales by employing the generalized riccati transformation we establish some new oscillation criteria for second order delay dynamic equations on time scales the obtained results essentially improve the well know oscillation results for half linear dynamic equations such as kamenev type and philos type oscillation criteria some interesting examples are given to illustrate the versatility of our results

a cs decomposition for orthogonal matrices with application to eigenvalue computation abstract we show that a schur form of a real orthogonal matrix can be obtained from a full cs decomposition based on this fact a cs decomposition based orthogonal eigenvalue method is developed we also describe an algorithm for orthogonal similarity transformation of an orthogonal matrix to a condensed product form and an algorithm for full cs decomposition the latter uses mixed shifted and zero shift iterations for high accuracy numerical examples are presented

convergence of the shifted qr algorithm for unitary hessenberg matrices this paper shows that for unitary hessenberg matrices the qr algorithm with an exceptional initial value modification of the wilkinson shift gives global convergence moreover the asymptotic rate of convergence is at least cubic higher than that which can be shown to be quadratic only for hermitian tridiagonal matrices under no further assumption a general mixed shift strategy with global convergence and cubic rates is also presented

on tridiagonal matrices unitarily equivalent to normal matrices abstract in this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied it is well known that any matrix is unitarily equivalent to a tridiagonal matrix in case of a normal matrix the resulting tridiagonal inherits a strong relation between its super and subdiagonal elements the corresponding elements of the super and subdiagonal will have the same absolute value in this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied both an iterative reduction based on krylov sequences as a direct tridiagonalization procedure via householder transformations are reconsidered this equivalence transformation is then applied to the normal case and equality of the absolute value between the super and subdiagonals is proved self adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved properties when applying the reduction on symmetric skew symmetric hermitian skew hermitian and unitary matrices and their relations with e g complex symmetric and pseudo symmetric matrices are presented it is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the takagi factorization finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given

algorithms for the geronimus transformation for orthogonal polynomials on the unit circle abstract let l be a positive definite bilinear functional on the unit circle defined on p n the space of polynomials of degree at most n then its geronimus transformation l is defined by l p q l z p z z q z for all p q p n c given l there are infinitely many such l which can be described by a complex free parameter the hessenberg matrix that appears in the recurrence relations for orthogonal polynomials on the unit circle is unitary and can be factorized using its associated schur parameters recent results show that the unitary hessenberg matrices associated with l and l respectively are related by a qr step where all the matrices involved are of order n 1 for the analogue on the real line of this so called spectral transformation the tridiagonal jacobi matrices associated with the respective functionals are related by an lr step in this paper we derive algorithms that compute the new schur parameters after applying a geronimus transformation we present two forward algorithms and one backward algorithm the qr step between unitary hessenberg matrices plays a central role in the derivation of each of the algorithms where the main idea is to do the inverse of a qr step making use of the special structure of unitary hessenberg matrices all the algorithms are efficient and need only o n flops we present several numerical experiments to analyse the accuracy and to explain the behaviour of the algorithms

oscillation of second order damped dynamic equations on time scales abstract the study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations the general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so called time scale which may be an arbitrary closed subset of the reals this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained in this paper by employing the riccati transformation technique we will establish some oscillation criteria for second order linear and nonlinear dynamic equations with damping terms on a time scale t our results in the special case when t r and t n extend and improve some well known oscillation results for second order linear and nonlinear differential and difference equations and are essentially new on the time scales t h n h 0 t q n for q 1 t n 2 etc some examples are considered to illustrate our main results

oscillation criteria for second order nonlinear neutral delay dynamic equations in this paper we will establish some oscillation criteria for the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t here 0 is a quotient of odd positive integers with r t and p t real valued positive functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales so this paper initiates the study

oscillation of second order delay differential equations on time scales abstract by means of riccati transformation technique we establish some new oscillation criteria for a second order delay differential equation on time scales in terms of the coefficients

on investigating gmres convergence using unitary matrices for a given matrix a and right hand side b this paper investigates unitary matrices generating with some right hand sides c the same gmres residual norms as the pair a b we give characterizations of this class of unitary matrices and point out the relationship with krylov subspaces and krylov residual subspaces for the pair a b we investigate the eigenvalues of these unitary matrices in relation to the convergence behavior of gmres for the pair a b and describe the indispensable role of the eigenvector information we conclude with a formula for the gmres residual norms generated by a normal matrix b in terms of its eigenvalues and components of the right hand side c in the eigenvector basis

asymptotic behavior of solutions of a third order nonlinear dynamic equation on time scales in this paper we will establish some sufficient conditions which guarantee that every solution of the third order nonlinear dynamic equation c t a t x t q t f x t 0 t t0 oscillates or converges to zero

homotopy algorithm for symmetric eigenvalue problems the homotopy method can be used to solve eigenvalue eigenvector problems the purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results in practice it is rerely of any interest to compute all the eigenvalues the homotopy method having the order preserving property can provide any specific eigenvalue without calculating any other eigenvalues besides this advantage we note that the homotopy algorithm is to a large degree a parallel algorithm numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices

oscillatory behavior of second order half linear damped dynamic equations by refining the generalized riccati transformation technique we obtain new oscillation criteria for a class of second order half linear delay dynamic equations with damping on a time scale assumptions in our theorems are less restrictive these results complement and improve related contributions to the subject in particular the results obtained amend related results reported in the literature

a new shift of the ql algorithm for irreducible symmetric tridiagonal matrices a new shift in the ql algorithm for symmetric tridiagonal matrices is described the shift is a combination of the rayleigh quotient shift and wilkinson s shift it is shown that ql is globally convergent with this shift and that the asymptotic rate is always cubic

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

oscillation criteria of second order half linear dynamic equations on time scales in this paper by using the riccati transformation technique chain rule and inequalitya l lab l 1 l 1 b l 0 l 1 where a and b are positive constants we will establish some oscillation criteria for the second order half linear dynamic equation p t x d t c d q t x c t 0 t a b on time scales where c 1 is an odd positive integer our results not only unify the oscillation of half linear differential and half linear difference equations but can be applied on different types of time scales and improve some well known results in the difference equation case some examples are considered here to illustrate our main results

convergence of the unitary qr algorithm with a unimodular wilkinson shift in applying the qr algorithm to compute the eigenvalues of a unitary hessenberg matrix a projected wilkinson shift of unit modulus is proposed and proved to give global convergence with at least a quadratic asymptotic rate for the qr iteration experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence

darboux transformations for cmv matrices abstract we develop a theory of darboux transformations for cmv matrices canonical representations of the unitary operators in perfect analogy with their self adjoint version the darboux transformations of jacobi matrices they are equivalent to laurent polynomial modifications of the underlying measures we address other questions which emphasize the similarities between darboux transformations for jacobi and cmv matrices like their almost isospectrality or the relation that they establish between the corresponding orthogonal polynomials showing also that both transformations are connected by the szeg mapping nevertheless we uncover some features of the darboux transformations for cmv matrices which are in striking contrast with those of the jacobi case in particular when applied to cmv matrices the matrix realization of the inverse darboux transformations what we call darboux transformations with parameters leads to spurious solutions whose interpretation deserves future research such spurious solutions are neither unitary nor band matrices so darboux transformations for cmv matrices are much more subject to the subtleties of the algebra of infinite matrices than their jacobi counterparts a key role in our theory is played by the cholesky factorizations of infinite matrices actually the darboux transformations introduced in this paper are based on the cholesky factorizations of degree one hermitian laurent polynomials evaluated on cmv matrices these transformations are also generalized to higher degree laurent polynomials as well as to the extension of cmv matrices to quasi definite functionals what we call quasi cmv matrices furthermore we show that this cmv version of darboux transformations plays a role in integrable systems like the schur flows or the ablowitz ladik model which parallels that of darboux for jacobi matrices in the toda lattice

minimal surfaces in kahler surfaces and ricci curvature soit m une surface immergee dans une surface de kahler n soient p le nombre de points tangents complexes et q le nombre de points tangents anticomplexes denombres selon leur multiple cite on deduit une version locale de p q x m x ou x m est la caracteristique d euler de m et x la caracteristique d euler du fibre normal de m dans n

an algorithm for finding generalized eigenpairs of a symmetric definite matrix pencil abstract a new method for finding eigenpairs of any symmetric definite matrix pencil is proposed it is a modification of the well known rayleigh quotient iteration it circumvents a drawback of that method i e its nonconvergence or slow convergence for poor initial vectors the global convergence and cubic convergence rate of the new method are proved its efficiency is also demonstrated by numerical results

kamenev type oscillation criteria for second order nonlinear dynamic equations on time scales the purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation r t x t f t x g t 0 on an arbitrary time scale t where is a quotient of odd positive integers and r is a positive rd continuous function on t the function g t t satisfies g t t and limt g t and f c t r r we establish some new sufficient conditions such that the above equation is oscillatory by using generalized riccati transformation our results in the special cases when t r and t n involve and improve some oscillation results for second order differential and difference equations and when t hn t qn0 and t n2 our oscillation results are essentially new some examples illustrating the importance of our results are included

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

oscillation and asymptotic behavior of third order nonlinear retarded dynamic equations this paper is concerned with oscillation and asymptotic behavior of a class of third order nonlinear delay dynamic equations on an arbitrary time scale a new theorem is presented that improves a number of results reported in the literature examples are included to illustrate new results

on the convergence rate of the ql algorithm with wilkinson s shift abstract w hoffmann and b n parlett have mentioned that in the usual case the convergence rate of the ql algorithm with wilkinson s shift is at least cubic in this paper we present a sufficient condition for supercubic convergence it is still an open problem whether or not convergence is always cubic

rational qr iteration without inversion in this manuscript a new method will be presented for performing a qr iteration with a i a i 1 qr without explicit inversion of the factor a i 1 a qr method driven by a rational function is attractive since convergence can occur at both sides of the matrix each step of this new iteration consists of two substeps in the explicit version first an rq factorization of the initial matrix a i rq will be computed followed by a qr factorization of the matrix a i q h the factorization of a i q h can be computed in an intelligent manner exploiting properties of the already known rq factorization of a i the similarity transformation yielding the qr step is defined by the unitary factor q in the qr factorization of the transformed matrix a i q h examples will be given illustrating how to efficiently compute the factorization for some specific classes of matrices the novelties of this approach with respect to these matrix classes will be discussed

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

the heat flow in nonlinear hodge theory we study the nonlinear hodge systemd 0 and 2 0 for an exterior form on a compact oriented riemannian manifold m where q is a given positive function the solutions are called harmonic forms they are the stationary points on cohomology classes of the functional e me 2 dm with e q q 2 the codifferential of a form is defined as 1 with 2 we evolve a given closed form 0 by the nonlinear heat flow system d for a time dependent exterior form x t on m this system is the differential of the normalized gradient flow u for e with 0 du under a technical assumption on the function 2 q q q we show that the nonlinear heat flow system d with initial condition 0 0 has a unique solution for all times which converges to a harmonic form in the cohomology class of 0 this yields a nonlinear hodge theorem that every cohomology class of m has a unique harmonic representative

unitary rank structured matrices in this paper we describe how one can represent a unitary rank structured matrix in an efficient way as a product of elementary unitary or givens transformations we also provide some basic operations for manipulating the representation such as the transition to zero creating form the transition to a unitary givens weight representation as well as an internal pull through process of the two branches of the representation finally we characterize how to determine the shift correction term to the rank structure and we provide some applications to this result

oscillation of nonlinear dynamic equations on time scales by means of riccati transformation techniques we establish some new oscillation criteria for the second order nonlinear dynamic equation p t x d t d q t f x s t 0 email protected a b on time scales in terms of the coefficients and the graininess function we illustrate our results with a nontrivial example

on the oscillation of third order neutral delay dynamic equations on time scales han et al z han t li s sun c zhang oscillation behavior of third order neutral emden fowler delay dynamic equations on time scales adv differential equations 2010 2010 article id 586312 23 pages gave some criteria for the oscillation and asymptotic behavior of solutions of the third order neutral delay dynamic equation r t x t a t x t t d d d p t x c d t 0 on a time scale t where c 0 is the quotient of odd positive integers in this paper the authors establish some new criteria for the oscillation of this equation that improve and unify those of han et al

eigenvalue computation for unitary rank structured matrices in this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps first we perform a reduction of the given matrix into hessenberg form next we compute the eigenvalues of this resulting hessenberg matrix via an implicit qr algorithm along the way we explain how the knowledge of a certain shift correction term to the structure can be used to speed up the qr algorithm for unitary hessenberg matrices and how this observation was implicitly used in a paper due to william b gragg we also treat an analogue of this observation in the hermitian tridiagonal case

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

oscillation of second order nonlinear neutral dynamic equations on time scales abstract this paper is concerned with the oscillation of second order nonlinear neutral dynamic equations of the form r t y t p t y t f t y t 0 on a time scale t by using a generalized riccati technique and integral averaging techniques we establish new oscillation results which handle some cases not covered by known criteria

compression of unitary rank structured matrices to cmv like shape with an application to polynomial rootfinding this paper is concerned with the reduction of a unitary matrix u to cmv like shape a lanczos type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the hermitian part of u i e of the matrix u u h by elaborating on the lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable if u is rank structured then the same property holds for its hermitian part and therefore the block tridiagonalization process can be performed using the rank structured matrix technology with reduced complexity our interest in the cmv like reduction is motivated by the unitary and almost unitary eigenvalue problem in this respect finally we discuss the application of the cmv like reduction for the design of fast companion eigensolvers based on the customary qr iteration

oscillation of second order half linear delay dynamic equations with damping on time scales by using the generalized riccati transformation and the inequality technique we establish a few new oscillation criterions for certain second order half linear delay dynamic equations with damping on a time scale our results extend and improve some known results but also unify the oscillation of the second order half linear delay differential equation with damping and the second order half linear delay difference equation with damping

oscillations of delay differential equations with variable coefficients consider the delay differential equation t p t x t 0 where p t c t0 r and is a positive constant we show that every solution of this equation oscillates if tt p t dt 1 e for sufficiently large t and t0 p t exp tt p s ds 1 e 1 dt

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

convergence of the tridiagonal qr algorithm by use of the three term recurrence relation an elementary and constructive proof is given for the global convergence of the symmetric tridiagonal qr algorithm with wilkinson s shift it is further illustrated why the asymptotic rate of convergence is essentially cubic as has long been observed in numerical experiments a general mixed shift strategy with global convergence and cubic rate is also presented

a variational characterization of complex submanifolds in this note we generalize our results in arezzo and sun reine angew math doi 10 1515 crelle 2013 0097 2012 to integer p currents of any degree we prove that if the mass of a current as a functional of the ambient metric has a critical or stable point in some special directions then the current is complex this holds for any dimension and codimension we also study a natural functional on the space of currents representing a fixed homology class closely related to the first derivative of the mass in our new approach detecting the deviation of a surface from being holomorphic

oscillation theorems for second order half linear delay dynamic equations with damping on time scales by using the generalized riccati transformation and the inequality technique we establish a few new oscillation criteria for certain second order half linear delay dynamic equations with damping on a time scale our results extend and improve some known results but also unify the oscillation of the second order half linear delay differential equation with damping and the second order half linear delay difference equation with damping

on a disconjugacy criterion for second order dynamic equations on time scales in this paper a sufficient condition for disconjugacy of second order differential equations on time scales is given for this purpose a mean value result on time scales is employed

a unitary hessenberg qr based algorithm via semiseparable matrices in this paper we present a novel method for solving the unitary hessenberg eigenvalue problem in the first phase an algorithm is designed to transform the unitary matrix into a diagonal plus semiseparable form then we rely on our earlier adaptation of the qr algorithm to solve the dpss eigenvalue problem in a fast and robust way exploiting the structure of the problem enables us to yield a quadratic time using a linear memory space nonetheless the algorithm remains robust and converges as fast as the customary qr algorithm numerical experiments confirm the effectiveness and the robustness of our approach

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

monotonicity oscillations and stability of a solution to a nonlinear equation modelling the capillary rise abstract in this paper we analyse a singular second order nonlinear ode which models the capillary rise of a fluid inside a tubular column we prove global existence uniqueness and find several approximations along with the asymptotic behaviour of the solution moreover we are able to find a critical value of the nondimensional parameter for which the solution exhibits a transition in its behaviour from being monotone to oscillatory this is an analytical rigorous proof of the experimentally and numerically confirmed phenomenon

oscillatory behavior of third order nonlinear delay dynamic equations on time scales in this paper a class of third order nonlinear delay dynamic equations on time scales is studied by using the generalized riccati transformation and the integral averaging technique three new sufficient conditions which ensure that every solution is oscillatory or converges to zero are established the results obtained essentially generalize and improve earlier ones

kamenev type and interval oscillation criteria for second order linear differential equations on a measure chain abstract we establish kamenev type criteria and interval criteria for oscillation of the second order scalar differential equation p t x t q t x t 0 on a measure chain our results cover those for differential equations and provide new oscillation criteria for difference equations several examples are given to show the significance of the results

interval oscillation criteria for second order forced delay dynamic equations with mixed nonlinearities interval oscillation criteria are established for second order forced delay dynamic equations on time scales containing mixed nonlinearities of the form r t f a x d t d p 0 t f a x t 0 t i 1np i t f b i x t i t e t t t 0 t where t is a time scale t 0 t a fixed number t 0 t is a time scale interval f u u 1u the functions r p i e t 0 t r are right dense continuous with r 0 nondecreasing t k t t are nondecreasing right dense continuous with t k t t lim t t k t and the exponents satisfy b 1 b m a b m 1 b n 0 all results are new even for t r and t z analogous results for related advance type equations are also given as well as extended delay and advance equations the theory can be applied to second order dynamic equations regardless of the choice of delta or nabla derivatives two examples are provided to illustrate one of the theorems

oscillation criteria for second order nonlinear delay dynamic equations abstract in this paper we consider the second order nonlinear delay dynamic equation r t x t p t f x t 0 on a time scale t by employing the generalized riccati technique we will establish some new sufficient conditions which ensure that every solution oscillates or converges to zero the obtained results improve the well known oscillation results for dynamic equations and include as special cases the oscillation results for differential equations some applications and examples are considered to illustrate the main results

stability of harmonic maps of kahler manifolds soit m n une application harmonique stable d une surface de riemann fermee dans un espace symetrique hermitien simple compact alors est holomorphe

the restarted qr algorithm for eigenvalue computation of structured matrices the qr algorithm is a popular numerical method for the computation of eigenvalues of matrices all eigenvalues of a general n n upper hessenberg matrix typically can be computed in o n3 arithmetic floating point operations using o n2 storage locations when the upper hessenberg matrix is hermitian or unitary then it can be represented by o n parameters and there are variants of the qr algorithm that reduce the operation count for computing all eigenvalues to o n2 arithmetic floating point operations and the storage requirement to o n locations however for many structured matrices that can be represented with o n storage locations available implementations of the qr algorithm require o n3 arithmetic floating point operations and o n2 storage locations to determine all eigenvalues this note shows that for some of the latter matrices the operation count can be reduced to o n2 arithmetic floating point operations and the memory requirement to o n storage locations by periodically restarting the qr algorithm

oscillation of second order nonlinear delay dynamic equations on time scales by using the generalized riccati transformation and the inequality technique we establish one new oscillation criterion for the second order nonlinear delay dynamic equations on time scales our results not only extend and improve some known theorems but also unify the oscillation of the second order nonlinear delay differential equation and the second order nonlinear delay difference equation on time scales

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved