stability of spherically symmetric steady states in galactic dynamics against general perturbations certain steady states of the vlasov poisson system can be characterized as minimizers of an energy casimir functional and this fact implies a non linear stability property of such steady states in previous investigations by y guo and g rein stability was obtained only with respect to spherically symmetric perturbations in the present investigation we show how to remove this non physical restriction

on high relative accuracy of the kogbetliantz method the relative accuracy of the kogbetliantz method for computing the singular value decomposition of real triangular matrices is considered sharp relative error bounds for the computed singular values are derived the results are obtained by estimating the norms of scaled perturbation matrices arising from errors generated in one step and one batch of commuting transformations perturbation matrices are scaled either from the left or right side by the inverse of the diagonal matrix

time analyticity and backward uniqueness of weak solutions of the navier stokes equations of multidimensional compressible flow abstract we prove that solutions of the navier stokes equations of three dimensional compressible flow restricted to fluid particle trajectories can be extended as analytic functions of complex time as consequences we derive backward uniqueness of solutions as well as sharp rates of smoothing for higher order lagrangean time derivatives the solutions under consideration are in a reasonably broad regularity class corresponding to small energy initial data with a small degree of regularity the latter being required for conversion to the lagrangean coordinate system in which the analysis is carried out

accuracy of two svd algorithms for 2 2 triangular matrices a new algorithm for the accurate computation of the singular value decomposition of 2x2 triangular matrices is proposed the algorithm is based on voevodin formulas sharp accuracy bounds are derived by using a subtle error analysis which tracks the signs of the errors of intermediate quantities and does not neglect the non linear parts of the errors the analysis is fine tuned for the case of almost diagonal matrices the same analysis is also used to analyze the errors for the xlasv2 computational routine of lapack the error estimates of the new algorithm compare favorably to those of the lapack routine

global existence and blow up phenomena for nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions abstract in this paper we consider nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions we establish respectively the conditions on nonlinearities to guarantee that u x t exists globally or blows up at some finite time if blow up occurs we obtain upper and lower bounds of the blow up time

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

accuracy of the kogbetliantz method for scaled diagonally dominant triangular matrices the paper proves that kogbetliantz method computes all singular values of a scaled diagonally dominant triangular matrix which can be well scaled from both sides symmetrically to high relative accuracy special attention is paid to deriving sharp accuracy bounds for one step one batch and one sweep of the method by a simple numerical test it is shown that the methods based on bidiagonalization are generally not accurate on that class of well behaved matrices

unstable and stable galaxy models to determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the vlasov theory for galaxy dynamics in this article we study the stability of isotropic spherical symmetric galaxy models f0 e for which the distribution function f0 depends on the particle energy e only in the first part of the article we derive the first sufficient criterion for linear instability of f0 e f0 e is linearly unstable if the second order operator a 0 equiv delta 4 pi int f 0 prime e i mathcal p dv has a negative direction where mathcal p is the projection onto the function space g e l l being the angular momentum see the explicit formulae 29 and 28 in the second part of the article we prove that for the important king model the corresponding a0 is positive definite such a positivity leads to the nonlinear stability of the king model under all spherically symmetric perturbations

nonlinear stability of circular vortex patches this paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the euler equations of incompressible fluids this is achieved by establishing a relative variational principle in terms of either energy or angular momentum thus we exploit and extend arnold s idea in 1965 1969 to a nonsmooth setting as well

on the convergence of cyclic jacobi like processes abstract we study a complex column and row cyclic jacobi like process of the form a k 1 u k a k v k f k k 1 where u k v k are unitary plane matrices f k is a sequence converging to diagonal form and a 1 is an arbitrary n n matrix a simple necessary and sufficient condition for the convergence of the sequence a k to diagonal form is obtained an estimate for the norms of certain cyclic jacobi operators is also obtained

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

on nonlinear stability of isotropic models in stellar dynamics consider an aggregation of mass particles in space which attract each other according to newton s law of attraction this system can be described by distribution functions satisfying the vlasov equation and the poisson laws we obtain the nonlinear stability of certain stationary states including those obeying modified emden s laws a priori estimates of the energy casimir functions around stationary states are established for distribution functions for which the l 5 3 norms of the density functions are uniformly bounded a uniform bound on the kinetic energy of the system readily implies that these norms of the density functions are indeed uniformly bounded in this way we prove nonlinear stability

euler evolution for singular initial data and vortex theory we study the evolution of a two dimensional incompressible ideal fluid in a case in which the vorticity is concentrated in small disjoint regions of the physical space we prove for short times a connection between this evolution and the vortex model

on the quadratic convergence of kogbetliantz s algorithm for computing the singular value decomposition abstract it is shown that the cyclic kogbetliantz algorithm ultimately converges quadratically when no pathologically close singular values are present

backward solutions to neumann and dirichlet problems of heat conduction equation applying energy methods we show the backward uniqueness in time for solutions to neumann and dirichlet problems of heat conduction equations and give reasonable physical interpretation for our conclusion

full block j jacobi method for hermitian matrices abstract the paper considers convergence accuracy and efficiency of a block j jacobi method the method is a proper blas 3 generalization of the known method of veselic for computing the hyperbolic singular value decomposition of rectangular matrices at each step the proposed algorithm diagonalizes the block pivot submatrix the convergence is proved for cyclic strategies which are weakly equivalent to the row cyclic strategy the relative accuracy is proved under the standard conditions numerical tests show improved performance with respect to the block oriented generalization of the original method of veselic combined with the hermitian indefinite factorization the proposed method becomes accurate and efficient eigensolver for hermitian indefinite matrices

orbital stability of spherical galactic models we consider the three dimensional gravitational vlasov poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters a well known conjecture is the stability of spherical models which are nonincreasing radially symmetric steady states solutions this conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by antonov in 1961 in a previous work we derived the stability of anisotropic models under it spherically symmetric perturbations using fundamental monotonicity properties of the hamiltonian under suitable generalized symmetric rearrangements first observed in the physics litterature in this work we show how this approach combined with a it new generalized antonov type coercivity property implies the orbital stability of spherical models under general perturbations

on efficient implementations of kogbetliantz s algorithm for computing the singular value decomposition in this paper we compare several implementations of kogbetliantz s algorithm for computing the svd on sequential as well as on parallel machines comparisons are based on timings and on operation counts the numerical accuracy of the different methods is also analyzed

steady states in galactic dynamics by minimizing the energy for the vlasov poisson system under a constraint guo and rein have constructed a large class of isotropic spherically symmetric steady states they have shown that an isolated minimizer is automatically dynamically stable under general i e not necessarily symmetric perturbations the main result of this work is to remove the assumption that the minimizer must be isolated so minimizers are stable even if they are not isolated it is also shown that the lagrange multipliers associated with all minimizers have the same value finally an example where two distinct minimizers exist is studied numerically

on kogbetliantz s svd algorithm in the presence of clusters we consider matrices with off diagonal blocks of small norm and derive tight bounds for the approximation of their singular values by those of their diagonal blocks these results are used to show that triangular matrices with clusters of singular values must possess a principal submatrix of nearly diagonal form from the latter we then derive results pertaining to the quadratic convergence of kogbetliantz s algorithm for computing the svd in the presence of clusters

triangular processor array for computing singular values a triangular processor array for computing the singular values of an m n m n matrix is proposed a jacobi type algorithm is used to first triangularize the given matrix and then diagonalize the resultant triangular form the requirements are 14n2 o n processors and o m ns time where s denotes the number of sweeps the triangular array can be extended to a rectangular one with 12mn o m processors for the computational of singular vectors

on sharp quadratic convergence bounds for the serial jacobi methods using a new technique we derive sharp quadratic convergence bounds for the serial symmetric and svd jacobi methods for the symmetric jacobi method we consider the cases of well and poorely separated eigenvalues our result implies the result proposed but not correctly proved by van kempen it also extends the well known result of wilkinson to the case of multiple eigenvalues

linear convergence of the row cyclic jacobi and kogbetliantz methods linear convergence of the row cyclic jacobi and kogbetliantz methods can be guaranteed if certain constraints concerning the angles of rotations are implemented unlike the results of forsythe and henrici convergence can be achieved without under rotations