discrete dynamical systems associated with root systems of indefinite type a geometric charactrization of the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but which exhibits chaotic behavior is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be a realization of a cremona isometry on the picard group of the surface it is also shown that the group of cremona isometries is isomorphic to an extended weyl group of indefinite type a method to construct the mappings associated with some root systems of indefinite type is also presented

deautonomizing integrable non qrt mappings we examine a class of previously derived integrable mappings which do not belong to the qrt family and show that they can be extended to non autonomous forms without loss of the integrable character we derive more non qrt integrable mappings obtain their non autonomous forms and show how they can be integrated

elliptic discrete painlev equations we present explicit forms of discrete painleve equations in which the independent variable and the parameter enter through the arguments of elliptic functions these equations have eight degrees of freedom and the geometry of their transformations is described by the affine weyl group e 1 8

limits and degeneracies of discrete painleve equations a sequel abstract we present the discrete systems which result from the discrete painleve equations q p vi and d p v associated to the affine weyl group e 7 1 two different procedures limits and degeneracies are used giving rise to a host of new discrete painleve equations but also to some equations which are integrable through linearisation

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

10e9 solution to the elliptic painleve equation a function formalism for sakai s elliptic painleve equation is presented this establishes the equivalence between the two formulations by sakai and by ohta ramani grammaticos we also give a simple geometric description of the elliptic painleve equation as a non autonomous deformation of the addition formula on elliptic curves by using these formulations we construct a particular solution of the elliptic painleve equation expressed in terms of the elliptic hypergeometric function 10e9

array shape calibration using sources in unknown locations a maximum likelihood approach sensor location uncertainty can severely degrade the performance of direction finding systems an iterative maximum likelihood method for simultaneously estimating directions of arrival doa and sensor locations is developed to alleviate this problem the case of nondisjoint sources i e sources observed in the same frequency cell and at the same time is emphasized the algorithm converges to the global maximum of the likelihood function if the initial conditions are sufficiently good numerical examples are presented illustrating the performance of the proposed technique a distinctive feature of the algorithm is its ability to locate the sensors accurately without deploying calibration sources at known locations

fiber dependent deautonomization of integrable 2d mappings and discrete painleve equations it is well known that two dimensional mappings preserving a rational elliptic fibration like the quispel roberts thompson mappings can be deautonomized to discrete painleve equations however the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated in this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber starting from a single autonomous mapping but varying the type of a chosen fiber we obtain different types of discrete painleve equations using this deautonomization procedure we also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the picard group and some additional geometric data this technique allows us to obtain factorized expressions of discrete painleve equations including the elliptic case further by imposing certain restrictions on such non autonomous mappings we obtain new and simple elliptic difference painleve equations including examples whose symmetry groups do not appear explicitly in sakai s classification

on almost strong approximation for algebraic groups abstract if g is a simply connected reductive group defined over a number field k and is the set of all infinite places of k then g has strong approximation with respect to if and only if the archimedean part of any k simple component of the adele group g a is non compact using affine bruhat tits buildings we formulate an almost strong approximation asap for groups of compact type extending the version treated in j s hsia m jochner invent math 129 1997 471 487 the validity of asap for g k is proved for all classical groups of compact type whose tits indices over k are not 2 a n d with d 3 application to genera of integral forms similar to gross notion of z models b gross invent math 124 1996 263 279 is given with attendant results on integral representations of positive definite quadratic hermitian or skew hermitian forms

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

integrable mappings via rational elliptic surfaces we present a geometric description of the qrt map which is an integrable mapping introduced by quispel roberts and thompson in terms of the addition formula of a rational elliptic surface by this formulation we classify all the cases when the qrt map is periodic and show that its period is 2 3 4 5 or 6 a generalization of the qrt map which acts birationally on a pencil of k3 surfaces or calabi yau manifolds is also presented

elliptic and multiplicative discrete painleve equations from the affine weyl group e8 pursuing our study of discrete painlev equations associated with the affine weyl group we present our results on multiplicative and elliptic equations our method is based on the trihomographic representation of discrete painlev equations which was shown to be equally rich as the more customary representations the relation of equations to ones associated with affine weyl groups situated in the degeneration cascade of is also presented since it constitutes a precious aid for the derivation of the systems

on almost strong approximation for some exceptional groups if g is a simply connected semisimple group defined over a number field k and is the set of all infinite places of k then g has strong approximation with respect to if and only if the archimedean part of any k simple component of the adele group ga is non compact using the affine bruhat tits building the authors of w k chan j hsia on almost strong approximation of algebraic groups j algebra 254 2002 441 formulated an almost strong approximation property asap for groups of compact type and they proved that asap holds for all classical groups of compact type whose tits indices over k are not 2an d with d 3 in this paper we show that asap holds for groups of types 3 6d4 g2 f4 e7 or e8

discrete painleve equations associated with the affine weyl group e8 we examine a family of discrete systems which define analogues of the painleve equations and which are related to the affine weyl group a domain hardly explored to date using the method of deautonomization we derive new integrable mappings the coefficients of which involve periodic functions with periods 14 24 and 30 our method is based on a novel representation of these discrete painleve equations dubbed trihomographic which we show to be generic

almost strong approximations for definite quadratic spaces a fundamental result in the arithmetic of integral quadratic forms is the strong approximation theorem for the spin group spin of a non degenerate inde nite quadratic form in three or more variables in practice one invokes the corresponding version for the kernel o0 of the spinor norm function this result is due to the works of eichler and kneser ei kn in the 1950s and has ever since played a central role in the proofs of a number of important theorems for both inde nite as well as de nite quadratic forms on the other hand it is well known that such a strong approximation cannot possibly hold for i the group so of a inde nite quadratic form as it is not simply connected and ii for the group spin of a de nite quadratic form as spin1 is compact the purpose of this paper is to show that there exists a version of an almost strong approximation theorem for de nite quadratic forms one forom and one foro m m 3 and show that they too have serious applications in the arithmetic theory e g in the asymptotic distributions of classes in a genus and in the representation theory of de nite forms it is anticipated that such almost strong approximations would also extend to other classical groups and will be treated in a future article we give a brief outline of the contents of this paper terminology and notations are generally those from om for simplicity as well as convenience we shall consider here only positive de nite integral z lattices a key step in the proof of the strong approximation theorem for o0 is the following result on inde nite representations with approximation property invent math 129 471 487 1997

on the limits of discrete painleve equations associated with the affine weyl group e8 we study the discrete painleve equations that can be obtained as limits from the equations associated with the affine weyl group e8 1 we obtain equations associated with the groups e7 1 and e6 1 as well as linearisable systems in the e7 1 and e6 1 cases we obtain several new discrete painleve equations along with equations which can be related to the ones already known the same is true for linearisable systems in the case of new linearisable mappings we present their explicit linearisation

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

how to find the conserved quantities of nonlinear discrete equations a method of finding conserved quantities of nonlinear ordinary difference equations is briefly discussed the method is based on the hypothesis that the conserved quantities are expressed by a ratio of two polynomials of dependent variables we obtain the conserved quantities of the discrete equation of motion of an anharmonic oscillator and a nonlinear difference equation of third order

simultaneous computation of hecke operators abstract we present a new method for computing hecke operators acting on spaces of algebraic modular forms based on an idea of eichler s the method significantly improves the complexity of computing operators that can be obtained this way and in addition computes two operators simultaneously we show that in certain cases the approach can be used to attain the action of the full hecke algebra with respect to a hyperspecial subgroup and use it to compute hecke eigenforms for compact forms of symplectic groups

from the continuous pv to discrete painleve equations we study the discrete transformations that are associated with the auto backlund of the continuous pv equation we show that several two parameter discrete painleve equations can be obtained as contiguity relations of pv among them we find the asymmetric d pii equation which is a well known form of discrete piii the relation between the ternary pi previously obtained through the discrete dressing approach and pv is also established a new discrete painleve equation is also derived

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure