structure arising from induction and jacquet modules of representations of classical p adic groups jacquet modules are very useful in the study of parabolically induced representations of reductive groups over a p adic field f we shall assume char f 2 in this paper it is very hard to describe explicitly the structure of jacquet modules of parabolically induced representations particularly in the most interesting cases there exists a description of factors of certain filtrations on them that description was done by j bernstein and a v zelevinsky in bz2 and by w casselman in c in the case of general linear groups the functor of parabolic induction and the jacquet functor induce a structure of z graded hopf algebra on the sum r of grothendieck groups of categories of smooth representations of gl n f s of finite length n 0 z1 the multiplication m r r r is defined using parabolic induction while the comultiplication m r r r is defined in terms of jacquet modules the most interesting part of the structure is the property that m r r r is a ring homomorphism hopf axiom this enables one to compute composition series of parabolically induced representations in a very simple way it is interesting to note that the existence of this strong structure did not have serious impact on the development of the representation theory of gl n f one of the reasons for that may lay in the fact that for gl n f there existed a very powerful theory of gelfandkazhdan derivatives and the main results of z1 were obtained using them nevertheless a v zelevinsky showed in z2 that some interesting parts of the representation theory of gl n over a finite field can be obtained as a structure theory of such hopf algebra defined in this setting besides that one of the main tools for the study of representation theory of gl n over a central division f algebra in t1 is such hopf algebra structure in this situation are not available gelfand kazhdan derivatives it is natural to ask does some structure of this kind exist for other simple split classical p adic groups since levi factors of classical groups are isomorphic to direct products of general linear groups and smaller groups from the same series one can expect for such structure to have some relation with r if it exists in the third section we define the direct sum of grothendieck groups r s which corresponds either to the series sp n f n 0 or so 2n 1 f n 0 in a similar way as r was defined for general linear groups the action of r on r s is defined using the parabolic induction in this way r s becomes z graded r module we can make r s a z graded comodule over r the comodule structure map r s r r s is again defined using the jacquet modules similarly as in the case of gl n f it is not hard to see that r s is not a hopf module over r see remark 7 3 in this paper we

korovkin tests approximation and ergodic theory we consider sequences of s k n x t k n matrices a n f with a block structure spectrally distributed as an l 1 p variate s x t matrix valued function f and for any n we suppose that a n is a linear and positive operator for every fixed n we approximate the matrix a n f in a suitable linear space m n of s k n x t k n matrices by minimizing the frobenius norm of a n f x n when x n ranges over m n the minimizer x n is denoted by p k n a n f we show that only a simple korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts 1 the sequence p k n a n f is distributed as f 2 the sequence a n f p k n a n f is distributed as the constant function 0 i e is spectrally clustered at zero the first result is an ergodic one which can be used for solving numerical approximation theory problems the second has a natural interpretation in the theory of the preconditioning associated to cg like algorithms

linear independence of intertwining operators abstract for an irreducible admissible representation of a connected reductive p adic group we consider standard intertwining operators holomorphic at zero using algebraic methods connected with the structure of linear algebraic groups we control supports of particularly chosen functions from the induced space we prove linear independence of standard intertwining operators this is used to extend the definition of the r group from a square integrable representation to its aubert involution

the best circulant preconditioners for hermitian toeplitz systems ii the multiple zero case in 10 14 circulant type preconditioners have been proposed for ill conditioned hermitian toeplitz systems that are generated by nonnegative continuous functions with a zero of even order the proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions it was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems in 10 14 it was claimed that this result can be the case where the generating functions have multiple zeros the main aim of this paper is to give a complete convergence proof of the method in 10 14 for this class of generating functions

the spectral approximation of multiplication operators via asymptotic structured linear algebra abstract a multiplication operator on a hilbert space may be approximated with finite sections by choosing an orthonormal basis of the hilbert space multiplication operators with nonzero symbols defined on l 2 spaces of functions are never compact and then such approximations cannot converge in the norm topology instead we consider how well the spectra of the finite sections approximate the spectrum of the multiplication operator whose expression is simply given by the essential range of the symbol i e the multiplier we discuss the case of real orthogonal polynomial bases and the relations with the classical fourier basis whose choice leads to the well studied toeplitz case indeed the asymptotic approximation of the spectrum by the spectra of the associated toeplitz sections is possible only under precise geometric assumptions on the range of the symbol conversely the use of circulant approximations leads to constructive algorithms with o n log n complexity n number of sections working in general and generalizable to the separable multivariate and matrix valued cases as well

spectral behavior of matrix sequences and discretized boundary value problems abstract in this paper we provide theoretical tools for dealing with the spectral properties of general sequences of matrices of increasing dimension more specifically we give a unified treatment of notions such as distribution equal distribution localization equal localization clustering and sub clustering as a case study we consider the matrix sequences arising from the finite difference fd discretization of elliptic and semielliptic boundary value problems bvps the spectral analysis is then extended to toeplitz based preconditioned matrix sequences with special attention to the case where the coefficients of the differential operator are not regular belong to l 1 and to the case of multidimensional problems the related clustering properties allow the establishment of some ergodic formulas for the eigenvalues of the preconditioned matrices

singular values and eigenvalues of non hermitian block toeplitz matrices in this paper we are concerned with the analysis of the distribution and localization of the singular values of toeplitz matrices ftn f g generated by a p variate lebesgue integrable matrix valued function f q p c hk q we prove that the union of the essential ranges of the singular values of f is a proper weak cluster for the whole set of the singular values of ftn f g by showing that the number of outliers is strongly depending on the regularity features of the underlying function f in particular if f is continuous or from the krein algebra and p 1 then the cluster is proper other results concerning the extreme spectral behavior of ftn f g second order ergodic formulas and localization of eigenvalues of preconditioned matrices ft 1 n g tn f g are presented some examples of applications to the preconditioning of these results are also discussed c 1999 elsevier science b v all rights reserved msc 15a12 15a18 65f10

preconditioning strategies for hermitian indefinite toeplitz linear systems in this paper we propose and analyze preconditioning strategies for hermitian indefinite linear systems by using indefinite preconditioners under very elementary assumptions we show that the eigenvalues are real moreover in the case of multilevel toeplitz structures we prove distributional and localization results these techniques used in connection with the cg gmres bicgstab and qmr algorithms allow us to solve in an optimal way the corresponding linear systems a wide numerical experimentation confirms the efficiency of the proposed procedures

eigenvalues of hermitian toeplitz matrices with smooth simple loop symbols abstract the paper presents higher order asymptotic formulas for the eigenvalues of large hermitian toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line the formulas are established not only for the extreme eigenvalues but also for the inner eigenvalues the results extend and make more precise existing results which so far pertain to banded matrices or to matrices with infinitely differentiable symbols also given is a fixed point equation for the eigenvalues which may be solved numerically by an iteration method

komleva type expansions and asymptotics for linear operators abstract in this paper we prove asymptotic komleva type expansions valid for sequences of linear operators t n approximating the identity in supremum norm over the space of the continuous functions in particular under suitable mild conditions on the sequence t n we obtain rational expansions for t n f that are of special interest in a numerical analysis context as special cases of these results we find asymptotic expansions for exponential type and de la vallee poussin polynomial operators the case of the cesaro sums is discussed in connection with the komleva theory but the main asymptotic results are proved by using other tools coming from a context of structured linear algebra some numerical applications of the theoretical part are then discussed

on the extreme eigenvalues of hermitian block toeplitz matrices abstract we are concerned with the behavior of the minimum maximum eigenvalue 0 n n n of an n 1 n 1 hermitian toeplitz matrix tn where is an integrable real valued function kac murdoch and szego widom parter and r h chan obtained that 0 n min o 1 n2k in the case where c2k at least locally and inf has a zero of order 2k we obtain the same result under the second hypothesis alone moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices proving that the rate of convergence of 0 n to inf depends only on the order not necessarily even or integer or finite of the zero of inf with the help of this tool we derive an absolute lower bound for the minimal eigenvalues of toeplitz matrices generated by nonnegative l1 functions and also an upper bound for the associated euclidean condition numbers finally these results are extended to the case of hermitian block toeplitz matrices with toeplitz blocks generated by a bivariate integrable function

the extension of the concept of the generating function to a class of preconditioned toeplitz matrices abstract it is well known that the generating function f l 1 of a class of hermitian toeplitz matrices a n f describes very precisely the spectrum of each matrix of the class recently this theory has been extended in various senses in three directions the non hermitian case the block case and the case of preconditioned toeplitz matrices here by using some results of the first and the second extension we deduce new properties of distribution of the spectrum of preconditioned matrices and we define the concept of generating function for these matrices moreover we discuss the consequences of these results on the analysis of convergence of preconditioned conjugate gradient methods previously defined in the literature finally we perform some numerical experiments which confirm the theoretical idea

variable projection methods for approximate greatest common divisor computations we consider the problem of finding for a given n tuple of polynomials real or complex the closest n tuple that has a common divisor of degree at least d extended weighted euclidean seminorm of the coefficients is used as a measure of closeness two equivalent representations of the problem are considered i direct parameterization over the common divisors and quotients image representation and ii sylvester low rank approximation kernel representation we use the duality between least squares and least norm problems to show that i and ii are closely related to mosaic hankel low rank approximation this allows us to apply to the approximate common divisor problem recent results on complexity and accuracy of computations for mosaic hankel low rank approximation we develop optimization methods based on the variable projection principle both for image and kernel representation these methods have linear complexity in the degrees of the polynomials for small and large d we provide a software implementation of the developed methods which is based on a software package for structured low rank approximation

asymptotic eigenvalue distribution of block toeplitz matrices and application to blind simo channel identification szego s 1984 theorem states that the asymptotic behavior of the eigenvalues of a hermitian toeplitz matrix is linked to the fourier transform of its entries this result was later extended to block toeplitz matrices i e covariance matrices of multivariate stationary processes the present work gives a new proof of szego s theorem applied to block toeplitz matrices we focus on a particular class of toeplitz matrices those corresponding to covariance matrices of single input multiple output simo channels they satisfy some factorization properties that lead to a simpler form of szego s theorem and allow one to deduce results on the asymptotic behavior of the lowest nonzero eigenvalue for which an upper bound is developed and expressed in terms of the subchannels frequency responses this bound is interpreted in the context of blind channel identification using second order algorithms and more particularly in the case of band limited channels

branching random walk in random environment phase transitions for local and global growth rates let n be the infinite particle system on whose evolution is as follows at each unit of time each particle independently is replaced by a new generation the size of a new generation descending from a particle at sitex has distributionf x and each of its members independently jumps to sitex 1 with probability 1 h 2 h 0 1 the sequence f x is i i d with uniformly bounded second moment and is kept fixed during the evolution the initial configuration 0 is shift invariant and ergodic two quantities are considered

c g preconditioning for toeplitz matrices abstract we consider the problem of solving a toeplitz system of equations by conjugate gradient method when a sequence of nested toeplitz matrices is associated to a function the spectral behaviour of the matrices involved is closely related to the analytical properties of the generating function thus it is possible to devise efficient preconditioning techniques by using various functional approximation strategies this approach leads to attractive results in the case of ill conditioned matrices for which a wide class of preconditioners are proposed

quenched sub exponential tail estimates for one dimensional random walk in random environment suppose that the integers are assigned i i d random variables x taking values in the unit interval which serve as an environment this environment defines a random walk x n called a rwre which when at x moves one step to the right with probability x and one step to the left with probability 1 x solomon 1975 determined the almost sure asymptotic speed v rate of escape of a rwre greven and den hollander 1994 have proved a large deviation principle for x n n conditional upon the environment with deterministic rate function for certain environment distributions where the drifts 2 x 1 can take both positive and negative values their rate function vanisheson an interval 0 v we find the rate of decay on this interval and prove it is a stretched exponential of appropriate exponent that is the absolute value of the log of the probability that the empirical mean x n n is smaller than v v 0 v behaves roughly like a fractional power of n the annealed estimates of dembo peres and zeitouni 1996 play a crucial role in the proof we also deal with the case of positive and zero drifts and prove there a quenched decay of the form

the best circulant preconditioners for hermitian toeplitz systems in this paper we propose a new family of circulant preconditioners for ill conditioned hermitian toeplitz systems a x b the preconditioners are constructed by convolving the generating function f of a with the generalized jackson kernels for an n by n toeplitz matrix a the construction of the preconditioners requires only the entries of a and does not require the explicit knowledge of f when f is a nonnegative continuous function with a zero of order 2p the condition number of a is known to grow as o n2p we show however that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p 1 outliers moreover the smallest eigenvalue is uniformly bounded away from zero hence the conjugate gradient method when applied to solving the preconditioned system converges linearly the total complexity of solving the system is therefore of o n log n operations in the case when f is positive we show that the convergence is superlinear numerical results are included to illustrate the effectiveness of our new circulant preconditioners

inside the eigenvalues of certain hermitian toeplitz band matrices while extreme eigenvalues of large hermitian toeplitz matrices have been studied in detail for a long time much less is known about individual inner eigenvalues this paper explores the behavior of the jth eigenvalue of an n by n banded hermitian toeplitz matrix as n tends to infinity and provides asymptotic formulas that are uniform in j for email protected email protected n the real valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum and then to decrease strictly back from the maximum to the minimum having nonzero second derivatives at the minimum and the maximum the results which are of interest in numerical analysis probability theory or statistical physics for example are illustrated and underpinned by numerical examples

two level toeplitz preconditioning approximation results for matrices and functions large 2 level toeplitz systems arise in a variety of applications see e g r h chan and m ng siam rev 38 1996 pp 427 482 for which efficient numerical methods for their solution are required some successful numerical techniques need the explicit knowledge of the generating function f of the considered system t n f rm x b an assumption that usually is not fulfilled in real applications in this paper we analyze and complete the procedure proposed in d noutsas s serra capizzano and p vassalos numer linear algebra appl 12 2005 pp 231 239 for the 2 level case in such a way from the knowledge of the coefficients of t n f we determine optimal preconditioning strategies for the solution of our systems finally some numerical experiments are performed and discussed in connection with our theoretical analysis

preconditioners for ill conditioned toeplitz systems constructed from positive kernels in this paper we are interested in the iterative solution of ill conditioned toeplitz systems generated by continuous nonnegative real valued functions f with a finite number of zeros we construct new w circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f kn with a suitable positive reproducing kernel kn by the restriction to positive kernels we obtain positive definite preconditioners moreover if f has only zeros of even order le 2s then we can prove that the property int pi pi t 2k k n t mbox d t le c n 2k k 0 hspace 1 5pt ldots s of the kernel is necessary and sufficient to ensure the convergence of the pcg method in a number of iteration steps independent of the dimension n of the system our theoretical results were confirmed by numerical tests

a law of large numbers for random walks in random environment we derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of kalikow the approach is based on the existence of a renewal structure under an assumption of transience in the direction l this extends to a multidimensional context previous work of kesten our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

precise large deviation estimates for a one dimensional random walk in a random environment suppose that the integers are assigned i i d random variables f xg taking values in the interval 1 2 1 which serve as an environment thisenvironmentdenesarandomwalk fxkg calledarwre which when at x moves one step to the right with probability x and one step to the left withprobability1 x solomon 1975 determinedthealmost sureasymp totic speed rate of escape of a rwre in a more general set up dembo peres and zeitouni 1996 following earlier work by greven and den hol lander 1994 on the quenched case have computed rough tail asymptotics for the empirical mean of the annealed rwre they conjectured the form of the rate function in a full ldp we prove in this paper their conjecture the proof is based on a icoarse graining schemei together with comparison techniques

on new examples of ballistic random walks in random environment in this article we show that random walks in random environment on zd d 3 with transition probabilities which are e perturbations of the simple random walk and such that the expectation of the local drift has size bigger than e with 52 when d 3 3 when d 4 fulfill the condition t introduced by sznitman prob theory related fields 2002 122 509 544 when e is small as a result these walks satisfy a law of large numbers with nondegenerate limiting velocity a central limit theorem and several large deviation controls in particular this provides examples of ballistic random walks in random environment which do not satisfy kalikow s condition in the terminology of sznitman and zerner ann probab 1999 27 1851 1869 an important tool in this work is the effective criterion of sznitman

korovkin theorems and linear positive gram matrix algebra approximations of toeplitz matrices abstract in this paper we are concerned with the approximation of toeplitz matrices generated by continuous 2 periodic functions f i r with i for this purpose we define a class of matrix algebras related to suitable choices of gram functions where we look for good preconditioners in particular we construct these preconditioners through linear operators and linear positive operators lpos approximating in some sense the function f then by making use of some matrix versions 39 42 41 of the korovkin and weierstrass theorems we analyze the convergence features of old and new preconditioners finally among the given results are adapted in order to deal with l1 generating functions and the related preconditioners are compared with the ones devised by using band toeplitz matrices 6 36

preconditioned hss methods for the solution of non hermitian positive definite linear systems and applications to the discrete convection diffusion equation we study the role of preconditioning strategies recently developed for coercive problems in connection with a two step iterative method based on the hermitian skew hermitian splitting hss of the coefficient matrix proposed by bai golub and ng for the solution of nonsymmetric linear systems whose real part is coercive as a model problem we consider finite differences fd matrix sequences an a p n discretizing the elliptic convection diffusion problem with being a plurirectangle of rd with a x being a uniformly positive function and p x denoting the reynolds function here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes more precisely in connection with preconditioned hss gmres like methods we consider the preconditioning sequence pn a n pn a dn1 2 a an 1 0 dn1 2 a where dn a is the suitably scaled main diagonal of an a 0 if a x is positive and regular enough then the preconditioned sequence shows a strong clustering at unity so that the sequence pn a n turns out to be a superlinear preconditioning sequence for an a 0 n where an a 0 represents a good approximation of re an a p namely the real part of an a p the computational interest is due to the fact that the preconditioned hss method has a convergence behavior depending on the spectral properties of pn 1 a re an a p n pn 1 a an a 0 n therefore the solution of a linear system with coefficient matrix an a p is reduced to computations involving diagonals and to the use of fast poisson solvers for an 1 0 n some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques

intermittent estimation for gaussian processes let xn n 0 be a stationary real valued gaussian time series we estimate the conditional expectation e xn 1 x0 xn from a growing number of observations x0 xn in a pointwise consistent way along a sequence of stopping times

approximation of nonnegative functions by means of exponentiated trigonometric polynomials we consider the problem of approximating a nonnegative function from the knowledge of its first fourier coefficients here we analyze a method introduced heuristically in a paper by borwein and huang siam j opt 5 1995 68 99 where it is shown how to construct cheaply a trigonometric or algebraic polynomial whose exponential is close in some sense to the considered function in this note we prove that approximations given by borwein and huang s method in the trigonometric case can be related to a nonlinear constrained optimization problem and their convergence can be easily proved under mild hypotheses as a consequence of known results in approximation theory and spectral properties of toeplitz matrices moreover they allow to obtain an improved convergence theorem for best entropy approximations

large deviations for random walks in a random environment we prove two large deviation principles for random walks in a random environment one for quenched walks in general ergodic environments and the other for averaged walks in a product environment 2003 wiley periodicals inc

error analysis of the tau method dependence of the error on the degree and on the length of the interval of approximation abstract in this paper we discuss the dependence of tau method approximations on i the degree of approximant n and ii the length of the interval of approximation h we shall show that the tau method parameters i i decay exponentially in terms of n and ii for a fixed n they decay in terms of h as h 2 n

an effective criterion for ballistic behavior of random walks in random environment we investigate multi dimensional random walks in random environment specifically we provide an effective criterion which can be checked by inspection of the environment in a finite box and implies a ballistic strong law of large numbers a central limit theorem and several large deviation controls with the help of this criterion we provide an example of a ballistic multi dimensional walk which does not fulfill the criterion introduced by kalikow in 4 the present work complements the results of 9 where a certain condition t was introduced and confirms the interest of this condition

intertwining operators and residues i weighted characters operateurs d entrelacement normalisation groupes reels groupes p adiques representations standard les distributions j m la distribution j m x residus changements de contour les espaces 7b h ac g f s et 7b f ac g f s l application m

tri diagonal preconditioner for pricing options the value of a contingent claim under a jump diffusion process satisfies a partial integro differential equation pide we localize and discretize this pide in space by the central difference formula and in time by the second order backward differentiation formula the resulting system t nx b in general is a nonsymmetric toeplitz system we then solve this system by the normalized preconditioned conjugate gradient method a tri diagonal preconditioner l n is considered we prove that under certain conditions all the eigenvalues of the normalized preconditioned matrix l n 1t n l n 1t n are clustered around one which implies a superlinear convergence rate numerical results exemplify our theoretical analysis

large deviation principle for random walk in a quenched random environment in the low speed regime we consider a one dimensional random walk x n n n in a random environment of zero or strictly positive drifts we establish a full large deviation principle for x n n of the correct order n log n 2 in the low speed regime valid for almost every environment this completes the large deviation picture obtained earlier by greven and den hollander and gantert and zeitouni in the case of zero and positive drifts the proof uses coarse graining along with concentration of measure techniques

a note on transience versus recurrence for a branching random walk in random environment we consider a branching random walk in random environment on d where particles perform independent simple random walks and branch according to a given offspring distribution at a random subset of sites whose density tends to zero at infinity given that initially one particle starts at the origin we identify the critical rate of decay of the density of the branching sites separating transience from recurrence i e the progeny hits the origin with probability 1 resp 1 we show that for d 3 there is a dichotomy in the critical rate of decay depending on whether the mean offspring at a branching site is above or below a certain value related to the return probability of the simple random walk the dichotomy marks a transition from local to global behavior in the progeny that hits the origin we also consider the situation where the branching sites occur in two or more types with different offspring distributions and show that the classification is more subtle due to a possible interplay between the types this note is part of a series of papers by the second author and various co authors investigating the problem of transience versus recurrence for random motions in random media

tail estimates for one dimensional random walk in random environment suppose that the integers are assigned i i d random variables x taking values in the unit interval which serve as an environment this environment defines a random walk x k called a rwre which when atx moves one step to the right with probability x and one step to the left with probability 1 x solomon 1975 determined the almost sure asymptotic speed rate of escape of a rwre for certain environment distributions where the drifts 2 x 1 can take both positive and negative values we show that the chance of the rwre deviating below this speed has a polynomial rate of decay and determine the exponent in this power law for environments which allow only positive and zero drifts we show that these large deviation probabilities decay like exp cn1 3 this differs sharply from the rates derived by greven and den hollander 1994 for large deviation probabilities conditioned on the environment as a by product we also provide precise tail and moment estimates for the total population size in a branching process with random environment

spectra of multilevel toeplitz matrices advanced theory via simple matrix relationships we consider the eigenvalue and singular value distributions for m level toeplitz matrices generated by a complex valued periodic function of m real variables we show that familiar formulations for l due to szeg and others can be preserved so long as f l1 and what is more with g weyl s definitions just a bit changed in contrast to other approaches the one we follow is based on simple matrix relationships

slowdown and neutral pockets for a random walk in random environment we consider a d dimensional random walk in random environment for which transition probabilities at each site are either neutral or present an effective drift pointing to the right we obtain large deviation estimates on the probability that the walk moves in a too slow ballistic fashion both under the annealed and quenched measures these estimates underline the key role of large neutral pockets of the medium in the occurrence of slowdowns of the walk

a variational approach to branching random walk in random environment this paper considers an infinite system of particles on the integers mathbb z that 1 step to the right with a random delay and 2 split or die along the way according to a random law depending on their position the exponential growth rate of the particle density is computed in the long time limit in the form of a variational formula that can be solved explicitly the result reveals two phase transitions associated with localization vs delocalization and survival vs extinction in addition the system exhibits an intermittency effect greven and den hollander considered the more difficult situation where the particles may step both to the left and right but the analysis of the phase diagram was less complete

superlinear pcg methods for symmetric toeplitz systems in this paper we deal with the solution by means of preconditioned conjugate gradient pcg methods of n x n symmetric toeplitz systems a n f x b with nonnegative generating function f here the function f is assumed to be continuous and strictly positive or is assumed to have isolated zeros of even order in the first case we use as preconditioner the natural and the optimal approximation of a n f proposed by bini and di benedetto and we prove that the related pcg method has a superlinear rate of convergence and a total arithmetic cost of o n log n ops under the second hypothesis we cannot guarantee that the natural matrix is positive definite while for the optimal we show that in the ill conditioned case this can be really a bad choice consequently we define a new matrix for preconditioning the given system then by applying the sherman morrison woodbury inversion formula to the preconditioned system we introduce a small constant number of subsidiary systems which can be solved again by means of the previous pcg method finally we perform some numerical experiments that show the effectiveness of the devised technique and the adherence with the theoretical analysis

optimal quasi optimal and superlinear band toeplitz preconditioners for asymptotically ill conditioned positive definite toeplitz systems in this paper we are concerned with the solution of n x n hermitian toeplitz systems with nonnegative generating functions f the preconditioned conjugate gradient pcg method with the well known circulant preconditioners fails in the case where f has zeros in this paper we consider as preconditioners band toeplitz matrices generated by trigonometric polynomials g of fixed degree l we use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n for each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy we obtain different estimates of l for which the total cost of the proposed pcg methods is optimal and the related rates of convergence are superlinear finally for the most economical strategy we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems

on multidimensional branching random walks in random environment we study branching random walks in random i i d environment in z d d geq 1 for this model the population size cannot decrease and a natural definition of recurrence is introduced we prove a dichotomy for recurrence transience depending only on the support of the environmental law we give sufficient conditions for recurrence and for transience in the recurrent case we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time

some theorems on linear positive operators and functionals and their applications abstract in this paper we prove two theorems concerning linear positive operators and functionals in a very general sense we show how these results can be used to derive new results in a wide range of applications from approximation theory to the numerical treatment of differential equations and to the numerical analysis connected with signals and images

convergence analysis of two grid methods for elliptic toeplitz and pdes matrix sequences summary the solution of large toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers 13 14 22 the technique was modified in 6 36 and a rigorous proof of convergence of the tgm two grid method was given in the special case where the generating function has only a zero at x 0 0 of order at most two here by extending the latter approach we perform a complete analysis of convergence of the tgm under the sole assumption that f is nonnegative and with a zero at x 0 0 of finite order an extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic pdes with nonconstant coefficients and of any order is then discussed

an ergodic theorem for classes of preconditioned matrices abstract we consider abstract classes of matrices a satisfying some structural conditions and in particular satisfying a crucial assumption about the asymptotic distribution of eigenvalues we prove a similar distribution property for classes of preconditioned matrices constructed by using representants of a as a particular case this result applies to preconditioned matrices coming from several important contexts finite differences and faedo ritz galerkin linear systems associated with elliptic and semielliptic boundary value problems very general hermitian toeplitz structures generated by multivariate ll functions this result answers in the positive some structural questions raised by tyrtyshnikov e tyrtyshnikov linear algebra appl 207 1994 225 249 and by the author s serra linear algebra appl 267 1997 139 161 s serra siam j numer anal in press in the toeplitz context

an approximate inverse preconditioner for toeplitz systems with multiple right hand sides abstract we are interested in preconditioning techniques for symmetric positive definite but ill conditioned toeplitz systems with multiple right hand sides tx b here the right hand sides are available simultaneously our main idea is to use an approximate inverse of t as a preconditioner for t the approximate inverse of t is generated by the inverse formula given by lv and huang x g lv t z huang a note on inversion of toeplitz matrices appl math lett 20 2007 1189 1193 the construction of the preconditioner is of low computational cost requires only the entries of t and does not require explicit knowledge of the generating function f of t we show that if given a proper parameter then the approximate inverse preconditioner approximates t 1 accurately and thus the spectra of these preconditioned matrices are clustered around 1 it follows that the conjugate gradient method converges very quickly when applied to solve the preconditioned systems numerical results are given to demonstrate the effectiveness of our preconditioner

strongly consistent online forecasting of centered gaussian processes an estimator e spl circ d sub n n of the conditional expectation e x sub n 1 x sub n x n d sub n 1 in a centered stationary and ergodic gaussian process x sub i sub i with absolutely summable wold coefficients is constructed on the basis of having observed x sub 1 x sub n for a suitable choice of the length dn spl rarr spl infin n spl rarr spl infin of the past covered by the conditional expectation it is established that e spl circ d sub n n e x sub n 1 x sub n x n d sub n 1 spl rarr 0 with probability 1 in addition sufficient conditions for e x sub n 1 x sub n x sub n 1 e x sub n 1 x sub n x n d sub n 1 spl rarr 0 to hold with probability 1 are given that is conditions under which e spl circ d sub n n can be used as a strongly consistent forecaster for e x sub n 1 x sub n x sub n 1

contact processes in several dimensions this paper deals primarily with the basic contact processes introduced and studied by harris 12t4 these are random evolutions on the state space s a l l subsets of z a zd the d dimensional integer lattice with extremely simple local dynamics namely if we think of the contact process as representing the spread of an infection t es denoting the set of infected sites at time t then x e it becomes healthy at exponential rate 1 while x z a t is infected at a rate proport ional to the number of sites neighboring x where infection is present the proportionali ty constant 2 is called the infection parameter contact processes are perhaps the simplest s valued markov processes which exhibit a critical phenomenon infection emanating from a single site dies out with probabili ty one for small positive 2 but has positive probability of surviving for all time when is large there is a critical value 2 d where the transition occurs principal objectives of analysis are the precise formulation of the critical phenomenon and detailed description of the ergodic behavior both below and above a c our recent survey articles 10 and i 6 provide introductions to contact processes and to the general field of interacting s valued systems respectively we will make constant use of notation and techniques from the surveys assuming that the reader is familiar with thern the article 10 describes in some detail the current state of knowledge concerning one dimensional contact processes the theory for d 1 is now fairly complete with two major exceptions 1 is not even determined to one decimal place 1 18 2 e there is an

lyapounov exponents and quenched large deviations for multidimensional random walk in random environment assign to the lattice sizes z z d i i d random 2d dimensional vectors z z e e 1 whose entries take values in the open unit interval and add up to one given a realization of this environment let x n n 0 be a markov chain on z d which when at z moves one step to its neighbor z e with transition probability z z e we derive a large deviation principle for x n n by means of a result similar to the shape theorem of first passage percolation and related models this result produces certain constants that are the analogue of the lyapounov exponents known from brownian motion in poissonian potential or random walk in random potential we follow a strategy similar to sznitman

block band toeplitz preconditioners derived from generating function approximations analysis and applications we are concerned with the study and the design of optimal preconditioners for ill conditioned toeplitz systems that arise from a priori known real valued nonnegative generating functions f x y having roots of even multiplicities our preconditioned matrix is constructed by using a trigonometric polynomial x y obtained from fourier kernel approximations or from the use of a proper interpolation scheme both of the above techniques produce a trigonometric polynomial x y which approximates the generating function f x y and hence the preconditioned matrix is forced to have clustered spectrum as x y is chosen to be a trigonometric polynomial the preconditioner is a block band toeplitz matrix with toeplitz blocks and therefore its inversion does not increase the total complexity of the pcg method preconditioning by block toeplitz matrices has been treated in the literature in several papers we compare our method with their results and we show the efficiency of our proposal through various numerical experiments

asymptotic behavior of the condition number of two level toeplitz matrix sequences in this note we give condition number estimates for two level toeplitz matrices generated by bivariate 2 periodic weakly sectorial symbols the largest class of symbols for which we can guarantee the invertibility of all the corresponding toeplitz matrices by using and extending tools previously developed in the one level context we show that the asymptotical ill conditioning is essentially related to the order of zeros of the symbol