value distribution of meromorphic solutions of certain difference painleve equations abstract the borel exceptional value and the exponents of convergence of poles zeros and fixed points of finite order transcendental meromorphic solutions for difference painleve i and ii equations are estimated and the forms of rational solutions of the difference painleve ii equation and the autonomous difference painleve i equation are also given it is also proved that the non autonomous difference painleve i equation has no rational solution

experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

an extension of the conjugate residual method to nonsymmetric linear systems the conjugate gradient cg method and the conjugate residual cr method are krylov subspace methods for solving symmetric positive definite linear systems to solve nonsymmetric linear systems the bi conjugate gradient bi cg method has been proposed as an extension of cg bi cg has attractive short term recurrences and it is the basis for the successful variants such as bi cgstab in this paper we extend cr to nonsymmetric linear systems with the aim of finding an alternative basic solver numerical experiments show that the resulting algorithm with short term recurrences often gives smoother convergence behavior than bi cg hence it may take the place of bi cg for the successful variants

growth and zeros of meromorphic solution of some linear difference equations abstract in this paper we study growth and zeros of linear difference equations p n z f z n p 1 z f z 1 p 0 z f z f z where f z p n z p 0 z are polynomials with f p n p 0 0 and satisfy deg p n p 0 max deg p j j 0 n 1 the corresponding homogeneous equation of the above equation is also investigated

qualitative behavior of non abelian toda like flows in this paper we analyze and compare the long term behavior of certain non abelian generalizations of the integrable toda lattice flow in particular we show that these flows exhibit generalized sorting behavior analogous to the sorting behavior of the original tridiagonal toda lattice we show that one of these generalizations the block double bracket equation is gradient with respect to a normal metric and exactly solvable this provides a new solvable case of the double bracket system

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

relationships between characteristic functions of δnf and f abstract in this paper we obtain the accurate relationships between the characteristic functions of a finite order meromorphic function f z and its forward differences n f z where f z has two borel exceptional values

topology of the isospectral real manifolds associated with the generalized toda lattices on semisimple lie algebras this paper concerns the topology of isospectral real manifolds of certain jacobi elements associated with real split semisimple lie algebras the manifolds are related to the compactified level sets of the generalized non periodic toda lattice equations defined on the semisimple lie algebras we then give a cellular decomposition and the associated chain complex of the manifold by introducing coloured dynkin diagrams which parametrize the cells in the decomposition we also discuss the morse chain complex of the manifold

on properties of meromorphic solutions for difference equations concerning gamma function abstract in this paper we mainly consider the properties of differences of meromorphic solutions for the difference equation a 1 z f z 1 a 0 z f z 0 concerning a gamma function where a 1 z and a 0 z are nonzero polynomials by these properties we deduce that a gamma function satisfies that for every n n n z z 0 n z has only n zeros z z z j z 1 j 0 1 n 1 z 1 z 1 n 1 z and 1 z have same zeros at most except n exceptional zeros where g denotes the order of growth of a meromorphic function g and g and g denote the exponents of convergence of fixed points and zeros of g respectively

the bicor and cors iterative algorithms for solving nonsymmetric linear systems we present two iterative algorithms for solving real nonsymmetric and complex non hermitian linear systems of equations and that were developed from variants of the nonsymmetric lanczos method in this paper we give the theoretical background of the two iterative methods and discuss their main computational aspects using a large number of numerical experiments we analyze their convergence properties and we also compare them with other popular nonsymmetric iterative solvers in use today

a comparative study of iterative solutions to linear systems arising in quantum mechanics this study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by sherry li of nersc numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed lanczos biconjugate a orthonormalization methods to other classic and popular iterative methods by the way experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence

bilinear trilinear forms and exact solution of certain fourth order integrable difference equations a systematic investigation of finding bilinear or trilinear representations of fourth order autonomous ordinary difference equation x n 4 f x n x n 1 x n 2 x n 3 or xn 4 f xn xn 1 xn 2 xn 3 is made as an illustration we consider fourth order symplectic integrable difference equations reported by capel and sahadevan physica a 289 86 2001 and derived their bilinear or trilinear forms also it is shown that the obtained bilinear representations admit exact solution of rational form

the second main theorem for small functions and related problems we shall establish the following three results in more general forms 1 the second main theorem for small functions let f be a meromorphic function on the complex plane c let a1 aq be distinct meromorphic functions on c assume that ai are small with respect to f i e t r a 0 corollary 1 here as usual in nevanlinna theory the terms t r f and n r ai f denote for the characteristic function and the truncated counting function respectively 2 application to functional equations let kc be the field of meromorphic functions on c for a function r 0 r put k c a kc t r a o r which is a subfield of kc then the following holds let f x y k c x y be a polynomial in two variables over k c assume that the curve f x y 0 over k c has genus greater than one if 1 2 kc satisfy the functional equation f 1 2 0 then both 1 and 2 are contained in k c corollary 2 3 height inequality for curves over function fields let k be a function field of one variable over c let x be a smooth projective curve over k let d x be a reduced divisor let l be an ample line bundle on x and let e 0 then we have hk kx d p n 1 k s d p dk p ehk l p oe 1 for all p x k d theorem 5 here the notations are introduced in v1 v3 see also section 9 our proof uses ahlfors theory of covering surfaces and the geometry of the moduli space of q pointed stable curves of genus 0

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

existence of finite order meromorphic solutions as a detector of integrability in difference equations abstract the existence of sufficiently many finite order in the sense of nevanlinna meromorphic solutions of a difference equation appears to be a good indicator of integrability it is shown that out of a large class of second order difference equations the only equation that can admit a sufficiently general finite order meromorphic solution is the difference painleve ii equation the proof given relies on estimates obtained by arguments related to singularity confinement the existence of meromorphic solutions of a general class of first order difference equations is also proven by a simple method based on banach s fixed point theorem

toda lattices with indefinite metric ii topology of the iso spectral manifolds abstract we consider the iso spectral real manifolds of tridiagonal hessenberg matrices with distinct real eigenvalues the manifolds are described by the iso spectral flows of indefinite toda lattice equations introduced by the authors physica d 91 1996 321 339 these toda lattices consist of 2n 1 different systems with hamiltonians h 1 2 k 1 n y k 2 k 1 n 1 s k s k 1 exp x k x k 1 where sk 1 which blow up in finite time except the case with all sksk 1 1 we compactify the manifolds by adding infinities according to the toda flows the resulting manifolds are shown to be nonorientable for n 2 and the symmetry group is the semi direct product of z 2 n 1 and the permutation group s n these properties identify themselves with small covers introduced by davis and januszkiewicz duke math j 62 1991 417 451 as a corollary of our construction we give a formula for the total number of zeros for a system of exponential polynomials generated as hankel determinants

ml k bicgstab a bicgstab variant based on multiple lanczos starting vectors we present a variant of the popular bicgstab method for solving nonsymmetric linear systems the method which we denote by ml k bicgstab is derived from a variant of the bicg method based on a lanczos process using multiple k 1 starting left lanczos vectors compared with the original bicgstab method our new method produces a residual polynomial which is of lower degree after the same number of steps but which also requires fewer matrix vector products to generate on average requiring only 1 1 k matvecs per step empirically it also seems to be more stable and more quickly convergent the new method can be implemented as a k term recurrence and can be viewed as a bridge connecting the arnoldi based fom gmres methods and the lanczos based bicgstab methods

an improved bi conjugate residual algorithm suitable for distributed parallel computing an improved bi conjugate residual bicr method ibicr method in brief for solving large sparse linear systems with unsymmetrical coefficient matrices was proposed for distributed parallel environments the method reduced two global synchronization points to one by reconstructing bicr method and all inner products per iteration were independent and communication time required for inner product can be overlapped efficiently with computation time of vector updates it combines the elements of numerical stability with the characters of design of parallel algorithms the cost is only a little increased computation performance and isoefficiency analysis shows that ibicr method has better parallelism and scalability than bicr method numerical experiments show that the parallel performance can be improved by a factor of about 2 we compared also ibicr with ibicg and bicr with bicg methods the results show that bicr and ibicr methods convergent at the same number of iteration and they convergent faster than bicg and ibicg methods respectively furthermore ibicr method conquers the vibration of residual norm of ibicg method

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

bicgcr2 a new extension of conjugate residual method for solving non hermitian linear systems abstract in the present paper we introduce a new extension of the conjugate residual cr method for solving non hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient bicg biconjugate residual bicr and biconjugate a orthogonal residual bicor methods the proposed krylov subspace method referred to as the bicgcr2 method is based on short term vector recurrences and is mathematically equivalent to both bicr and bicor we demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than bicg bicr and bicor hence it may be exploited for the development of new variants of non optimal krylov subspace methods

so p q toda systems abstract we define an integrable hamiltonian system of toda type associated with the real lie algebra s o p q as usual there exist a periodic and a non periodic version we construct using the root space two lax pair representations and the associated poisson tensors we prove liouville integrability and examine the multi hamiltonian structure the system is a projection of a canonical a n type toda lattice via a flaschka transformation it is also obtained via a complex change of variables from the classical toda lattice

a cocr method for solving complex symmetric linear systems the conjugate orthogonal conjugate gradient cocg method has been recognized as an attractive lanczos type krylov subspace method for solving complex symmetric linear systems however it sometimes shows irregular convergence behavior in practical applications in the present paper we propose a conjugate a orthogonal conjugate residual cocr method which can be regarded as an extension of the conjugate residual cr method numerical examples show that cocr often gives smoother convergence behavior than cocg

on the value distribution of some difference polynomials abstract in this paper we continue to investigate the value distribution of some difference polynomials which can be considered as difference counterparts of some classic results of hayman

closed form expressions for integrals of mkdv and sine gordon maps we present closed form expressions for approximately n integrals of 2n dimensional maps the maps are obtained by travelling wave reductions of the modified korteweg de vries equation and of the sine gordon equation respectively we provide the integrating factors corresponding to the integrals moreover we show how the integrals and the integrating factors relate to the staircase method

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

elementary exact calculations of degree growth and entropy for discrete equations second order discrete equations are studied over the field of rational functions c z where z is a variable not appearing in the equation the exact degree of each iterate as a function of z can b

singularity complexity and quasi integrability of rational mappings we investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models by exploiting their nature of birational transformations of projective spaces we give an algorithmic analysis of the structure of invariants of such mappings we discuss some characteristic conditions for their quasi integrability and in particular its links with their singularities in the 2 plane finally we describe some of their propertiesqua dynamical systems making contact with arnol d s notion of complexity and exemplify remarkable behaviours

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

isospectral flows on a class of finite dimensional jacobi matrices abstract we present a new matrix valued isospectral ordinary differential equation that asymptotically block diagonalizes n n zero diagonal jacobi matrices employed as its initial condition this o d e features a right hand side with a nested commutator of matrices and structurally resembles the double bracket o d e studied by r w brockett in 1991 we prove that its solutions converge asymptotically that the limit is block diagonal and above all that the limit matrix is defined uniquely as follows for n even a block diagonal matrix containing 2 2 blocks such that the super diagonal entries are sorted by strictly increasing absolute value furthermore the off diagonal entries in these 2 2 blocks have the same sign as the respective entries in the matrix employed as the initial condition for n odd there is one additional 1 1 block containing a zero that is the top left entry of the limit matrix the results presented here extend some early work of kac and van moerbeke

representations of mathrm sl n mathbb c and the toda lattice on montre que la methode dependante de la representation de van moerbeke et mumford pour transformer le flot des equations du reseau de toda en un flot lineaire sur le jacobien de la courbe spectrale dependant de la representation est independante de la representation de l algebre de lie utilisee dans celle de ces flots lineaires qui sont tous des images homomorphes localement injective d un autre sur des varietes abeliennes isogenes

idr s a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations we present idr s a new family of efficient short recurrence methods for large nonsymmetric systems of linear equations the new methods are based on the induced dimension reduction idr method proposed by sonneveld in 1980 idr s generates residuals that are forced to be in a sequence of nested subspaces although idr s behaves like an iterative method in exact arithmetic it computes the true solution using at most n n s matrix vector products with n the problem size and s the codimension of a fixed subspace we describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications our experiments show that idr s is competitive with or superior to most bi cg based methods and outperforms bi cgstab when s 1

lie algebraic aspects of the finite nonperiodic toda flows in this paper we review various aspects of nonperiodic full and tridiagonal toda flow theory in particular we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic toda equations we describe the structure of these systems on general semisimple lie algebras in the case of the full kostant asymmetric toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior we describe the relationship between the asymmetric toda flows and the symmetric and indefinite toda flows and show how one may conjugate from the full kostant toda flows to the full symmetric toda flows via a poisson map we describe briefly related toda systems such as nonabelian toda and the peakon flows

the level manifold of a generalized toda equation hierarchy the finite nonperiodic toda lattice equation induces a linear one parameter flow on a space of rational functions the level manifold of the toda equation hierarchy is shown to be a product of lines our main results establish a generalization of this toda hierarchy which will be called the cyclic toda hierarchy it is proved that the cyclic toda hierarchy is completely integrable and its level manifold is diffeomorphic to a disjoint union of cylinders

painleve analysis and normal forms theory nonlinear vector fields have two important types of singularities the fixed points in phase space and the time singularities in the complex plane the first singularities are locally analyzed via normal form theory whereas the second ones are studied by the painleve analysis in this paper normal form theory is used to describe the solutions around their complex time singularities to do so a transformation mapping the local series around the singularities to the local series around a fixed point of a new system is introduced regular normal form theory is then used in this new system it is shown that a vector field has the painleve property only if the associated system is locally linearizable around its fixed points a problem analogous to the classical problem of the center moreover the connection between partial and complete integrability and the structure of local series around both types of singularities are established a new proof of the convergence of the local psi series is given and an explicit method to prove the existence of finite time blow up manifold in phase space is presented

an atlas for tridiagonal isospectral manifolds abstract let t be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix with simple spectrum we introduce bidiagonal coordinates charts defined on open dense domains forming an explicit atlas for t in contrast to the standard inverse variables consisting of eigenvalues and norming constants every matrix in t now lies in the interior of some chart domain we provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics both for continuous and discrete time

some properties of meromorphic solutions of q difference equations abstract we investigate the growth of transcendental meromorphic solutions of some complex q difference equations and find lower bounds for nevanlinna lower order for meromorphic solutions of such equations we also obtain a q difference version of tumura clunie theorem

asymptotic hamiltonian dynamics the toda lattice the three wave interaction and the non holonomic chaplygin sleigh abstract in this paper we discuss asymptotic stability in energy preserving systems which have an almost poisson structure in particular we consider a class of poisson systems which includes the toda lattice in standard hamiltonian systems one of course does not expect asymptotic stability the key here is the structure of the phase space of the poisson or almost poisson systems and the nature of the equilibria our systems are in some sense generalizations of the integrable toda lattice system but are not integrable in general as a particular example we point out an interesting connection between three mechanical two degree of freedom systems that exhibit asymptotic stability two of them are classical hamiltonian systems while the third is a non holonomic system non holonomic systems are generally energy preserving but not hamiltonian but the system analyzed here turns out to have a phase space which is the union of hamiltonian ones we also discuss various higher dimensional examples

toda flows with infinitely many variables abstract the toda flow and related flows extend naturally to operators in hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

equations of camassa holm type and jacobi ellipsoidal coordinates we consider the integrable camassa holm ch equation on the line with positive initial data rapidly decaying at infinity on such a phase space we construct a one parameter family of integrable hierarchies that preserves the mixed spectrum of the associated string spectral problem this family includes the ch hierarchy we demonstrate that the constructed flows can be interpreted as hamiltonian flows on the space of weyl functions of the associated string spectral problem the corresponding poisson bracket is the atiyah hitchin bracket using an infinite dimensional version of the jacobi ellipsoidal coordinates we obtain a one parameter family of canonical coordinates linearizing the flows 2005 wiley periodicals inc

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

on the growth of logarithmic differences difference quotients and logarithmic derivatives of meromorphic functions a crucial ingredient in the recent discovery by ablowitz halburd herbst and korhonen 2000 2007 that a connection exists between discrete painleve equations and finite order nevanlinna theory is an estimate of the integrated average of log f z 1 f z on z r we obtained essentially the same estimate in our previous paper 2008 independent of halburd et al 2006 we continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences difference quotients and logarithmic derivatives for finite order meromorphic functions in addition to the potential applications of our new estimates in integrable systems they are also of independent interest in particular our findings show that there are marked differences between the growth of meromorphic functions with nevanlinna order less than and greater than one we have established a difference analogue of the classical wiman valiron type estimates for meromorphic functions with order less than one which allow us to prove that all entire solutions of linear difference equations with polynomial coefficients of order less than one must have positive rational order of growth we have also established that any entire solution to a first order algebraic difference equation with polynomial coefficients must have a positive order of growth which is a difference analogue of a classical result of polya

bicr variants of the hybrid bicg methods for solving linear systems with nonsymmetric matrices we propose bi conjugate residual bicr variants of the hybrid bi conjugate gradient bicg methods referred to as the hybrid bicr variants for solving linear systems with nonsymmetric coefficient matrices the recurrence formulas used to update an approximation and a residual vector are the same as those used in the corresponding hybrid bicg method but the recurrence coefficients are different they are determined so as to compute the coefficients of the residual polynomial of bicr from our experience it appears that the hybrid bicr variants often converge faster than their bicg counterpart numerical experiments show that our proposed hybrid bicr variants are more effective and less affected by rounding errors the factor in the loss of convergence speed is analyzed to clarify the difference of the convergence between our proposed hybrid bicr variants and the hybrid bicg methods

a generalized product type bicor method and its application in signal deconvolution for solving nonsymmetric linear systems we attempt to establish symmetric structures in nonsymmetric systems and handle them through the methods devised for symmetric cases a biconjugate a orthogonal residual method based on biconjugate a orthonormalization procedure has been proposed and nominated as bicor in y f jing t z huang y zhang l li g h cheng z g ren y duan t sogabe b carpentieri lanczos type variants of the cocr method for complex nonsymmetric linear systems j comput phys 228 2009 6376 6394 as many similar characteristics exist between bicor and bicg the strategies of improved variants of bicg such as cgs and bicgstab can be utilized to enhance the algorithm for bicor making use of the product of residual polynomials of bicor and other polynomials cors and bicorstab have been proposed along the same ideas of cgs and bicgstab respectively in the above mentioned paper in this paper a unified generalized framework of product type bicor which is epitomized by the product of residual polynomials and other polynomials is proposed numerical examples are selected from the blurring signal cases and the effect of the generalized product type bicor method is prominent in signal deconvolution

meromorphic solutions of difference painleve equations we show that under certain conditions some difference painleve equations have nontrivial meromorphic solutions in the whole complex plane these meromorphic solutions are obtained by analytic continuation of asymptotic solutions given in sectors of zero opening angle the existence of these asymptotic solutions is shown by using fundamental matrix solutions of the associated linear systems with double unit characteristic roots and with coefficients expressed by asymptotic series in fractional powers

nevanlinna theory for the q difference operator and meromorphic solutions of q difference equations it is shown that if f is a meromorphic function of order zero and q in mathbb c then begin equation label abstid m bigg r frac f qz f z bigg o t r f tag ddag end equation for all r on a set of logarithmic density 1 the remainder of the paper consists of applications of identity eqref abstid to the study of value distribution of zero order meromorphic functions and in particular zero order meromorphic solutions of q difference equations the results obtained include q shift analogues of the second main theorem of nevanlinna theory picard s theorem and clunie and mohon ko lemmas

parameterization by polytopes of intersections of orbits by conjugation abstract let s be an n n real symmetric matrix with spectral decomposition s q t q where q is an orthogonal matrix and is diagonal with simple spectrum 1 n also let o s and r s be the orbits by conjugation of s by respectively orthogonal matrices and upper triangular matrices with positive diagonal denote by f s the intersection o s r s we show that the map f f s r n taking s q t q to diag q q t is a smooth bijection onto its range p s the convex hull of some subset of the n points 1 2 n is apermutation we also find necessary and sufficient conditions for p s to have n vertices

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter