numerical range and functional calculus in hilbert space we prove an inequality related to polynomial functions of a square matrix involving the numerical range of the matrix we also show extensions valid for bounded and also unbounded operators in hilbert spaces which allow the development of a functional calculus

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

integrable mixing of an 1 type vertex models given a family of monodromy matrices t0 t1 tk 1 corresponding to integrable anisotropic vertex models of an 1 type 0 1 k 1 we built up a related mixed vertex model by means of gluing the lattices on which they are defined in such a way that integrability property is preserved the gluing process is implemented through one dimensional representations of rectangular quantum matrix algebras a rn 1 rn namely the gluing matrices algebraic bethe ansatz is applied on a pseudovacuum space with a selected basis and for each element of this basis it yields a set of nested bethe ansatz equations matching up to the ones corresponding to an am 1 quasiperiodic model with m equal to min zk rank

hopf bifurcation of viscous shock waves in compressible gas dynamics and mhd extending our previous results for artificial viscosity systems we show under suitable spectral hypotheses that shock wave solutions of compressible navier stokes and magnetohydrodynamics equations undergo hopf bifurcation to nearby time periodic solutions the main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in h s for data in h s we accomplish this by a novel energy estimate carried out in lagrangian coordinates interestingly this estimate is false in eulerian coordinates at the same time we greatly sharpen and simplify the analysis of the previous work

the numerical range and the spectrum of a product of two orthogonal projections the aim of this paper is to describe the closure of the numerical range of the product of two orthogonal projections in hilbert space as a closed convex hull of some explicit ellipses parametrized by points in the spectrum several improvements removing the closure of the numerical range of the operator using a parametrization after its eigenvalues are possible under additional assumptions an estimate of the least angular opening of a sector with vertex 1 containing the numerical range of a product of two orthogonal projections onto two subspaces is given in terms of the cosine of the friedrichs angle applications to the rate of convergence in the method of alternating projections and to the uncertainty principle in harmonic analysis are also discussed

principal angles between subspaces in an a based scalar product algorithms and perturbation estimates computation of principal angles between subspaces is important in many applications e g in statistics and information retrieval in statistics the angles are closely related to measures of dependency and covariance of random variables when applied to column spaces of matrices the principal angles describe canonical correlations of a matrix pair we highlight that all popular software codes for canonical correlations compute only cosine of principal angles thus making impossible because of round off errors finding small angles accurately we review a combination of sine and cosine based algorithms that provide accurate results for all angles we generalize the method to the computation of principal angles in an a based scalar product for a symmetric and positive definite matrix a we provide a comprehensive overview of interesting properties of principal angles we prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants numerical examples and a detailed description of our code are given

on the field of values of oblique projections abstract we highlight some properties of the field of values or numerical range w p of an oblique projector p on a hilbert space i e of an operator satisfying p 2 p if p is neither null nor the identity we present a direct proof showing that w p w i p i e the field of values of an oblique projection coincides with that of its complementary projection we also show that w p is an elliptical disk i e the set of points circumscribed by an ellipse with foci at 0 and 1 and eccentricity 1 p these two results combined provide a new proof of the identity p i p we discuss the influence of the minimal canonical angle between the range and the null space of p on the shape of w p in the finite dimensional case we show a relation between the eigenvalues of matrices related to these complementary projections and present a second proof to the fact that w p is an elliptical disk

directed bond percolation on a square lattice i analytical results a rigorous mathematical formulation of directed percolation is given in terms of matrix recursion equations the domany kinzel limit is rederived in a very simple way expansion around this limit yields surprisingly good results for the percolation probabilities already to first order

new conditions for non stagnation of minimal residual methods in the solution of large linear systems a condition guaranteeing that a minimal residual krylov subspace method makes some progress i e that it does not stagnate is that the symmetric part of the coefficient matrix be positive definite this condition results in a well established worst case bound for the convergence rate of the iterative method due to elman this bound has been extensively used e g when the linear system comes from discretized partial differential equations to show that the convergence of gmres is independent of the underlying mesh size in this paper we introduce more general non stagnation conditions which do not require the symmetric part of the coefficient matrix to be positive definite and that guarantee for example the non stagnation of restarted gmres for certain values of the restarting parameter

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

conditional stability of unstable viscous shocks abstract continuing a line of investigation initiated by texier and zumbrun on dynamics of viscous shock and detonation waves we show that a linearly unstable lax type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 2 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by howard mascia and zumbrun

on phase diagrams for directed percolation problems the authors propose a real space renormalisation group scheme for anisotropic directed bond percolation in two dimensions which differing from previous phenomenological or small cell real space renormalisation groups is sensitive to the existence of an infinite cluster along any lattice direction this enables them to explore new aspects of the phase diagram for the problem which were not accessible through existing treatments in particular the crossover exponent between one and two dimensional behaviour is shown to be exactly one for directed percolation numerical estimates are in agreement with known inequalities

pointwise green function bounds for shock profiles of systems with real viscosity following the pointwise semigroup approach of zh mz 1 we establish sharp pointwise green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic parabolic systems of conservation laws of dissipative type under the necessary assumptions z 1 z 3 z 4 of spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the connecting profile and hyperbolic stability of the corresponding ideal shock of the associated inviscid system with no additional assumptions on the structure or strength of the shock these bounds are used in a companion paper mz 2 to establish nonlinear stability of small amplitude lax shocks of symmetrizable hyperbolic parabolic systems

numerical testing of the stability of viscous shock waves a new theoretical evans function condition is used as the basis of a numerical test of viscous shock wave stability accuracy of the method is demonstrated through comparison against exact solutions a convergence study and evaluation of approximate error equations robustness is demonstrated by applying the method to waves for which no current analytic results apply highly nonlinear waves from the cubic model and strong shocks from gas dynamics an interesting aspect of the analysis is the need to incorporate features from the analytic evans function theory for purposes of numerical stability for example we find it necessary for numerical accuracy to solve odes on the space of wedge products

the refined inviscid stability condition and cellular instability of viscous shock waves abstract combining the work of serre and zumbrun benzoni gavage serre and zumbrun and texier and zumbrun we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite cross section duct the violation of the refined planar stability condition of zumbrun serre a viscous correction of the inviscid planar stability condition of majda more precisely we show for a model problem involving flow in a rectangular duct with artificial periodic boundary conditions that transition to multidimensional instability through violation of the refined stability condition of planar viscous shock waves on the whole space generically implies for a duct of sufficiently large cross section a cascade of hopf bifurcations involving more and more complicated cellular instabilities the refined condition is numerically calculable as described by benzoni gavage serre zumbrun

z 2 gauge model on fractal lattices hamiltonians with z 2 gauge symmetry are studied on a class of fractal lattices the lattices considered have fourier and hausdorf besicovitch dimensions strictly greater than 2 but they have a low degree of connectivity characterised by their connectivity index 2 it is proved that there is no phase transition at non zero temperatures

block triangular preconditioners for nonsymmetric saddle point problems field of values analysis a preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed the proposed preconditioner is of block triangular form the aim of this article is to show that a rigorous convergence analysis can be performed by using the field of values of the preconditioned linear system as an example a saddle point problem obtained from a mixed finite element discretization of the oseen equations is considered the convergence estimates obtained by using a field of values analysis are independent of the discretization parameter h several computational experiments supplement the theoretical results and illustrate the performance of the method

generic types and transitions in hyperbolic initial boundary value problems the stability of linear initial boundary value problems for hyperbolic systems with constant coefficients is linked to the zeros of the so called lopatinskii determinant depending on the location of these zeros problems may be either unstable strongly stable or weakly stable the first two classes are known to be open in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and or in the boundary condition here we show that a third open class exists which we call weakly stable of real type many examples of physical or mathematical interest depend on one or more parameters and the determination of the stability class as a function of these parameters usually needs an involved computation we simplify it by characterizing the transitions from one open class to another one these boundaries are easier to determine since they must solve some overdetermined algebraic system applications to the wave equation linear elasticity shock waves and phase boundaries in fluid mechanics are given

stability of small amplitude shock profiles of symmetric hyperbolic parabolic systems combining pointwise green s function bounds obtained in a companion paper 36 with earlier spectral stability results obtained in 16 we establish nonlinear orbital stability of small amplitude lax type viscous shock profiles for the class of dissipative symmetric hyperbolic parabolic systems identified by kawashima 20 notably including compressible navier stokes equations and the equations of magnetohydrodynamics obtaining sharp rates of decay in lp with respect to small l1 h3 perturbations 2 p our analysis extends and somewhat refines the approach introduced in 35 to treat stability of relaxation profiles 2004 wiley periodicals inc

general explicit descriptions for intertwining operators and direct rotations of two orthogonal projections abstract in this paper based on the block operator technique and operator spectral theory general explicit descriptions for intertwining operators and direct rotations of two orthogonal projections are established as a consequence kato s result kato 1996 14 is improved so are j avron r seiler and b simon s theorem 2 3 avron et al 1994 6 and c davis w m kahan s result davis and kahan 1970 11

galerkin eigenvector approximations how close are galerkin eigenvectors to the best approximation available out of the trial subspace under a variety of conditions the galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors both orthogonal galerkin and petrov galerkin methods are considered here with a special emphasis on nonselfadjoint problems thus extending earlier studies by chatelin babuska and osborn and knyazev consequences for the numerical treatment of elliptic pdes discretized either with finite element methods or with spectral methods are discussed new lower bounds to the sep of a pair of operators are developed as well

generalizations of the projection methods with application to sor theory for hermitian positive semidefinite linear systems ann n complex matrixb is calledparacontracting if b 2 1 and 0 x n i b bx 2 x 2 we show that a productb b k b k 1 b1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofb in terms of the subdominant singular values of theb i s and in terms of the angles between certain subspaces our results here extend earlier results due to halperin and due to smith solomon and wagner we also determine necessary and sufficient conditions forn numbers in the interval 0 1 to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product in the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the sor iteration matrix associated with ann n hermitian positive semidefinite matrixa none of whose diagonal entries vanish

a complete 2d stability analysis of fast mhd shocks in an ideal gas an algorithm of numerical testing of the uniform lopatinski condition for linearized stability problems for 1 shocks is suggested the algorithm is used for finding the domains of uniform stability neutral stability and instability of planar fast mhd shocks a complete stability analysis of fast mhd shock waves is first carried out in two space dimensions for the case of an ideal gas main results are given for the adiabatic constant 5 3 mono atomic gas that is most natural for the mhd model the cases 7 5 two atomic gas and 5 3 are briefly discussed not only the domains of instability and linear in the usual sense stability but also the domains of uniform stability for which a corresponding linearized stability problem satisfies the uniform lopatinski condition are numerically found for different given angles of inclination of the magnetic field behind the shock to the planar shock front as is known uniform linearized stability implies the nonlinear stability that is local existence of discontinuous shock front solutions of a quasilinear system of hyperbolic conservation laws

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

bounds on the eigenvalue range and on the field of values of non hermitian and indefinite finite element matrices in the early seventies fried formulated bounds on the spectrum of assembled hermitian positive semi definite finite element matrices using the extreme eigenvalues of the element matrices in this paper we will generalise these results by presenting bounds on the field of values the numerical radius and on the spectrum of general possibly complex matrices for both the standard and the generalised problem the bounds are cheap to compute involving operations with element matrices only we illustrate our results with an example from acoustics involving a complex non hermitian matrix as an application we show how our estimates can be used to derive an upper bound on the number of iterations needed to achieve a given residual reduction in the gmres algorithm for solving linear systems

variational analysis of the crouzeix ratio let w a denote the field of values numerical range of a matrix a for any polynomial p and matrix a define the crouzeix ratio to have numerator max left p zeta zeta in w a right max p w a and denominator vert p a vert 2 p a 2 crouzeix s 2004 conjecture postulates that the globally minimal value of the crouzeix ratio is 1 2 over all polynomials p of any degree and matrices a of any order we derive the subdifferential of this ratio at pairs p a for which the largest singular value of p a is simple in particular we show that at certain candidate minimizers p a the crouzeix ratio is clarke regular and satisfies a first order nonsmooth optimality condition and hence that its directional derivative is nonnegative there in every direction in polynomial matrix space we also show that pairs p a exist at which the crouzeix ratio is not regular

iterates of a product of conditional expectation operators abstract let f be a probability space and let t p 1 p 2 p d be a finite product of conditional expectations with respect to the sub algebras f 1 f 2 f d we show that for every f l p 1 p 2 the sequence t n f converges a e with lim n t n f e f f 1 f 2 f d a e

products of orthogonal projections and polar decompositions abstract we characterize the sets x of all products pq and y of all products pqp where p q run over all orthogonal projections and we solve the problems arg min p q p q z for z x or y we also determine the polar decompositions and moore penrose pseudoinverses of elements of x

theoretical and numerical structure for unstable one dimensional detonations the spatio temporal structure of unstable detonations in a single space dimension is studied through a combination of numerical and asymptotic methods a new high resolution numerical method for computing unstable detonations is developed this method combines the piecewise parabolic method ppm with conservative shock tracking and adaptive mesh refinement a new nonlinear asymptotic theory for the spatio temporal growth of instabilities is also developed this asymptotic theory involves a nonclassical hopf bifurcation because resonant acoustic scattering states with exponential growth in space cross the imaginary axis and become nonlinear eigenmodes in a complex free boundary problem for a nonlinear hyperbolic equation an interplay between the asymptotic theory and numerical simulation is used to elucidate the spatio temporal mechanisms of nonlinear stability near the transition boundary in particular a quantitative qualitative explanation is developed for the experimentally observed instabilities

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

field of values analysis of preconditioned iterative methods for nonsymmetric elliptic problems the convergence rate of krylov subspace methods for the solution of nonsymmetric systems of linear equations such as gmres or fom is studied bounds on the convergence rate are presented which are based on the smallest real part of the field of values of the coefficient matrix and of its inverse estimates for these quantities are available during the iteration from the underlying arnoldi process it is shown how these bounds can be used to study the convergence properties in particular the dependence on the mesh size and on the size of the skew symmetric part for preconditioners for finite element discretizations of nonsymmetric elliptic boundary value problems this is illustrated for the hierarchical basis and multilevel preconditioners which constitute popular preconditioning strategies for such problems

fields of values and iterative methods abstract the performance of an iterative scheme to solve x t x c t c n n c c n is often judged by spectral properties of t if t is not normal it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information to anticipate the progress of the iteration after a finite number of steps the knowledge of the eigenvalues alone is often useless in addition the spectrum of t may be highly sensitive to perturbations if t is not normal an iterative method which on the basis of some spectral information is predicted to converge rapidly for t may well diverge if t is slightly perturbed in practice the convergence of the iteration x m t x m 1 c is therefore frequently measured by some norm verbar t verbar rather than by the spectral radius t but apart from the fact that norms lead to error estimates which are often too pessimistic they cannot be used to analyze more general schemes such as e g the chebyshev iterative methods here we discuss another tool to analyze the behavior of an iterative method namely the field of values w t the collection of all rayleigh quotients of t w t contains the eigenvalues of t and the numerical radius t max z w t z defines a norm on c n n the field of values represents therefore an intermediate concept to judge an iterative scheme by it is related to the spectral approach but has also certain norm properties

asymmetric directed percolation on the square lattice the author considers directed percolation on the square lattice with probability ph pv for the horizontal vertical bonds to be unbroken for ph 1 epsilon epsilon small and pv sigma epsilon the percolating cluster is asymptotically within a cone phi phi phi the author calculates phi or and the fluctuations of the boundaries at phi phi or as power series in epsilon up to terms approximately epsilon 2 showing that the transverse spread of the percolating cluster is random walk like for any given ph the percolation threshold pv c epsilon defined by phi phi at pv pv c is also calculated