integral homology of real flag manifolds and loop spaces of symmetrical spaces abstract an algorithm is given for the computation of the integral homology groups of generalized real flag manifolds and of based loop spaces of compact symmetric spaces this extends earlier work of bott and samelson computing these invariants with z 2 coefficients the algorithm uses the root system of the underlying symmetric spaces to turn the computation into a relatively straightforward combinatorial calculation the proofs of the theorems use morse theory of the moment map as tool to understand the geometry of the spaces in question

the kostant toda equation and the hungry integrable systems abstract the toda equation which has applications in many fields is one of the most important integrable systems a generalization of the toda equation is the kostant toda equation whose lax representation is expressed using bidiagonal and lower hessenberg matrices the discrete toda equation which is a time discretization of the toda equation is extended to the so called discrete hungry toda dhtoda equation the lax representation for the dhtoda equation generates similarity transformations of lower hessenberg matrices although both have been considered a discrete analogue of the kostant toda equation or a continuous analogue of the dhtoda equation has yet to be presented in the literature in this paper we present a continuous analogue of the dhtoda equation and then relate this to the kostant toda equation by using a continuous analogue of the backlund transformation between the dhtoda equation and the discrete hungry lotka volterra system we relate the resulting continuous analogue to the continuous hungry lotka volterra system

on the distinguished eigenvalues of a cone preserving map abstract we generalize many known results on a nonnegative matrix concerning linear inequalities collatz wielandt sets and generalized eigenvectors to the setting of a matrix preserving a finite dimensional proper cone a simple cone theoretic proof is given for the nonnegative basis theorem for the algebraic eigenspace of a nonnegative matrix the result is also extended to a matrix preserving a polyhedral cone given proper cones k 1 and k 2 in different euclidean spaces a necessary and sufficient condition is also obtained for the existence of a nonzero matrix x which takes k 2 into k 1 and satisfies ax xb where a b are given matrices preserving k 1 and k 2 respectively this extends and answers a recent open question posed by hartwig

local spectral radii and collatz wielandt numbers of monic operator polynomials with nonnegative coefficients abstract operator polynomials l li l 1al 1 a1 a0 are considered where a0 al 1 are nonnegative operators in a banach space x with normal cone x for x x we define the local spectral radius rl x and the lower and upper collatz wielandt numbers r l x and r l x respectively of x with respect to l we characterize these quantities with the help of corresponding quantities with respect to the first companion operator belonging to l and the operator function s al 1 1al 2 l 1a0 many properties known in the linear case l 1 have generalizations to the case l 1 e g r l x r l x r l x is true for all x x from these local results we obtain results for the global spectral radius r l which were proved earlier under more restrictive conditions

tight estimates for convergence of some non stationary consensus algorithms the present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete time multi agent systems in the systems considered here both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time under the hypothesis that some spanning tree structure is preserved along time and that some non zero minimal weight of the information transfer along this tree is guaranteed an estimate of the contraction rate is given the latter is expressed explicitly as the spectral radius of some matrix depending upon the tree depth and the lower bounds on the weights

bounds for complete elliptic integrals of the second kind with applications in this paper we present some sharp bounds for complete elliptic integrals of the second kind as applications we establish several inequalities for the perimeter of an ellipse

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

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

complexity reduction for sign configurations through the kp ii equation and its information theoretic aspects we provide a combinatorial setting to explore the information content associated with the fulfillment of the kadomtsev petviashvili kp ii equation we start from a special family of solutions of kp ii namely tau functions of wronskian type a solution in this class can be expressed as a sum of exponentials and we look at combinations of signs signatures for the nonvanishing terms in this sum we prove a characterization of the signatures that return another solution of the kp ii equation they can be represented as choices of signs for columns and rows of a coefficient matrix so we recover a function satisfying the whole kp hierarchy from this single constraint the redundancy of this representation for different choices of the initial solution is investigated enumerative information theoretic and geometric aspects of this construction are discussed we provide a combinatorial setting to explore the information content associated with the fulfillment of the kadomtsev petviashvili kp ii equation we start from a special family of solutions of kp ii namely tau functions of wronskian type a solution in this class can be expressed as a sum of exponentials and we look at combinations of signs signatures for the nonvanishing terms in this sum we prove a characterization of the signatures that return another solution of the kp ii equation they can be represented as choices of signs for columns and rows of a coefficient matrix so we recover a function satisfying the whole kp hierarchy from this single constraint the redundancy of this representation for different choices of the initial solution is investigated enumerative information theoretic and geometric aspects of this construction are discussed

linear equations over cones and collatz wielandt numbers let k be a proper cone in r n let a be an n x n real matrix that satisfies ak k let b be a given vector of k and let be a given positive real number the following two linear equations are considered in this paper i i n a x b x k and ii a l n x b x k we obtain several equivalent conditions for the solvability of the first equation for the second equation we give an equivalent condition for its solvability in case when b a and we also find a necessary condition when b a and also when b a sufficiently close to b a where b a denotes the local spectral radius of a at b with fixed we also consider the questions of when the set a l n k k equals 0 or k and what the face of k generated by the set is then we derive some new results about local spectral radii and collatz wielandt sets or numbers associated with a cone preserving map and extend a known characterization of m matrices among z matrices in terms of alternating sequences

collatz wielandt numbers in general partially ordered spaces abstract it has been shown by forster and nagy that the convergence of the collatz wielandt numbers to bound the spectral radius of a k nonnegative linear operator t in a partially ordered bannach space essentially depends upon the starting vector of the iteration process in this paper necessary and sufficient conditions for convergence are presented under rather general hypotheses e g the emptiness of the interior of the order cone k is admitted we also present min sup and max inf characterizations of the spectral radius in the spirit of earlier work

classical and quantum mechanical systems of toda lattice type i solutions to the classical periodic and non periodic toda lattice type hamiltonian systems are expressed in terms of an iwasawa type factorization of a large lie group the scattering of these systems is determined in the non periodic case for the generalized periodic toda lattices a generalization of kostant s formula is obtained using standard representations of affine lie groups

characterization of critical values of branching random walks on weighted graphs through infinite type branching processes we study the branching random walk on weighted graphs site breeding and edge breeding branching random walks on graphs are seen as particular cases two kinds of survival can be identified a weak survival with positive probability there is at least one particle alive somewhere at any time and a strong survival with positive probability the colony survives by returning infinitely often to a fixed site the behavior of the process depends on the value of a certain parameter which controls the birth rates the threshold between survival and almost sure extinction is called critical value we describe the strong critical value in terms of a geometrical parameter of the graph we characterize the weak critical value and relate it to another geometrical parameter we prove that at the strong critical value the process dies out locally almost surely while at the weak critical value global survival and global extinction are both possible

a constructive approach to linear lyapunov functions for positive switched systems using collatz wielandt sets we establish the link between linear lyapunov functions for positive switched systems and corresponding collatz wielandt sets this leads to an algorithm to compute a linear lyapunov function whenever a lyapunov function exists

on the collatz wielandt sets associated with a cone preserving map abstract let k be a closed pointed full cone in r n in their treatment of perron frobenius theory for a linear map a preserving k by wielandt s approach barker and schneider introduced four sets namely 1 and 1 the collatz wielandt sets associated with a we determine the greatest lower bound and the least upper bound of these sets some known results for a nonnegative matrix relating its spectral radius to its upper and lower collatz wielandt number are extended to the setting of a cone preserving map applications of our results to the nonnegativity of solutions of linear inequalities associated with a nonnegative matrix are also considered

singular structure of toda lattices and cohomology of certain compact lie groups abstract we study the singularities blow ups of the toda lattice associated with a real split semisimple lie algebra g it turns out that the total number of blow up points along trajectories of the toda lattice is given by the number of points of a chevalley group k f q related to the maximal compact subgroup k of the group g with g lie g over the finite field f q here g is the langlands dual of g the blow ups of the toda lattice are given by the zero set of the functions for example the blow ups of the toda lattice of a type are determined by the zeros of the schur polynomials associated with rectangular young diagrams those schur polynomials are the functions for the nilpotent toda lattices then we conjecture that the number of blow ups is also given by the number of real roots of those schur polynomials for a specific variable we also discuss the case of periodic toda lattice in connection with the real cohomology of the flag manifold associated to an affine kac moody algebra

on approximating the arithmetic geometric mean and complete elliptic integral of the first kind abstract in the article we prove that the double inequalities 1 6 p 7 r p 5 p 6 r tanh 1 r 2 r k r 1 6 q 7 r q 5 q 6 r tanh 1 r 2 r q a 1 r 5 q 6 g 1 r a 1 r 6 q 7 g 1 r l 1 r a g m 1 r p a 1 r 5 p 6 g 1 r a 1 r 6 p 7 g 1 r l 1 r hold for all r 0 1 if and only if p 2 1 570796 and q 89 69 1 289855 where k r 0 2 1 r 2 sin 2 t 1 2 d t is the complete elliptic integral of the first kind tanh 1 r log 1 r 1 r 2 is the inverse hyperbolic tangent function r 1 r 2 and a 1 r 1 r 2 g 1 r r l 1 r r 1 log r and a g m 1 r are the arithmetic geometric logarithmic and gaussian arithmetic geometric means of 1 and r respectively

on the collatz wielandt numbers and the local spectral radius of a nonnegative matrix let t be a nonnegative linear continuous operator in a partially ordered banach space e and let x be a nonnegative nonzero vector in e we establish relations between the local spectral radius rt x and the upper and lower collatz wielandt numbers rt x and rt x examples will show that the situation in an infinite dimensional space fe can be much more complicated than in the classical case of rn with positive cone rn we also present in certain situations sufficient and or necessary conditions for both sequences rt tnx n and rt tnx n to converge to rt x the significance of some conditions in the infinite dimensional case is demonstrated and we make several comments on the earlier literature of the subject

the toda lattice dynkin diagrams singularities and abelian varieties the periodic l 1 particle toda lattices coresponding to extended dynkin diagrams have l 1 polynomial constants of the motion namely as many as there are dots in the dynkin diagram for most values of the constants their intersection defines as l dimensional affine invariant manifold which completes into a complex algebraic torus abelian variety by glueing on a divisor entirely specified by the extended dynkin diagram therefore the global geometry of the complex invariant tori such as polarization dimension of certain linear systems divisor equivalences etc is entirely given by the extended dynkin diagram in particular an explicit geometric description for the case of three particle toda lattices is given which describes the divisors their singularities and how the various divisors intersect

precise estimates for differences of the gaussian hypergeometric function abstract by showing certain combinations of the gaussian hypergeometric functions f a b a b 1 x c and f a b a b 1 x d to be monotone on 0 1 for given a b c d 0 a b and c d the authors study the problem of comparing these two functions they find sup 0 a f a b a b 1 x c f a b a b 1 x d for x 0 1 thus solving an open problem and a recent conjecture

schubert cells and representation theory let g be a linear real reductive group and let p be a parabolic subgroup the bruhat decomposition of g gives a cellular decomposition of the generalized flag manifold x g p originating in the work of schubert on grassmann manifolds this cellular decomposition was used by ehresmann eh to give a proof of the basis theorem for the integral cohomology of grassmannians since ehresmann it has been clear that for a generalized flag manifold x a more detailed understanding of this decomposition by generalized schubert cells would provide a determination of the integral co homology of x the aim of this article is to give a representation theoretic determination of the differentials in the schubert cell decomposition of x and thereby obtain a combinatorial description of the integral co homology of x our primary tool will be the infinite dimensional representation theory of g if g and p are complex groups elementary considerations show that all schubert cells define non zero integral homology classes and that none are torsion moreover the cohomology has the well known connection to finitedimensional representation theory ct ko1 ko2 bt for real groups g and p even in low dimensions torsion can be present in the homology of real flag manifolds thus the major new development in this paper is that the infinite dimensional representation theory of g detects which real schubert cells define integral in particular torsion classes in the co homology the innovation on the representation theoretic side that makes possible the consideration of co homology with integer coefficients is the geometric formulation of representation theory introduced by beilinson bernstein to compute homology we use a topological technique that applies to spaces with filtration let x p be the union of schubert cells whose dimension

monotonicity theorems and inequalities for the complete elliptic integrals we prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind k and e these results lead to sharp symmetrical bounds for k and e which improve recently discovered inequalities

close to convexity properties of gaussian hypergeometric functions abstract let a be the class of normalized analytic functions in the unit disk let z be either zf a b c z or c ab f a b c z 1 where f a b c z denotes the classical hypergeometric function the purpose of this paper is to study close to convexity and hence univalency of z in the unit disc more generally we find conditions on a b c and such that satisfies re e i 1 z z 0 for all z and for some real 1 2 1 2

sharp approximations for the complete elliptic integrals of the second kind by one parameter means abstract let e r be the complete elliptic integrals of the second kind and j p a b be the one parameter mean of two distinct positive numbers a and b defined by j p a b p p 1 a p 1 b p 1 a p b p for p 1 0 in this study using the recurrence method with the sign rule of a special power series we prove that the double inequality 7 22 j 2 2 1 r 2 e r j 2 2 1 r holds for r 1 r 2 0 1 where the upper bound is sharp while the lower bound is related to the archimedes approximation 22 7 for using the same method we also easily show that both of the functions r 2 2 e r j 7 4 1 r r 12 and r 2 1 r 10 1 1 2 e r 1 j 7 4 1 r are absolutely monotonic on 0 1 which greatly improves the known results moreover it should be emphasized that our method may be used to explore special functions related to hypergeometric functions

weight filtrations for induced representations of real reductive lie groups soit g un groupe matriciel reel semi simple connexe et p man la decomposition de langlands d un sous groupe parabolique de g etant donnee une representation admissible de m et un caractere e de a on peut former la representation induite normalisee ip e de g on decrit une g filtration de ip e avec des sous quotients semi simples en termes d une m filtration connue a priori de a sous quotients semi simples dans le cas de caractere infinitesimal entier