connections with exotic holonomy connections with exotic holonomy lorenz johannes schwachhofer wolfgang ziller supervisor in 1955 berger ber partially classified the possible irreducible holonomy representations of torsion free connections on the tangent bundle of a manifold however it was shown by bryant br2 that berger s list is incomplete connections whose holonomy is not contained on berger s list are called exotic we investigate examples of a certain 4 dimensional exotic holonomy representation of sl 2 r we show that connections with this holonomy are never complete give explicit descriptions of these connections on an open dense set and compute their group of symmetry finally we give strong restrictions for their existence on compact manifolds

generalized hermite polynomials associated with functions of parabolic cylinder associated hermite polynomials he n n z which generalize the usual scaled hermite polynomials he n z corresponding to n 0 are introduced for the purpose to represent the raising and lowering of the indices of functions of the parabolic cylinder d n z in finite integer steps properties of these polynomials such as recursion relations explicit representations and the differential equation are derived the generation of the associated hermite polynomials from the usual hermite polynomials by differential operators representable by means of the confluent hypergeometric function is given an application for the explicit calculation of the functions of the parabolic cylinder for negative integer indices is discussed other applications are visible for the investigation of the zeros of the functions of the parabolic cylinder

integral transforms and chebyshev like polynomials we use the integral representation method to derive the properties of chebyshev polynomials and of their generalized forms

homogeneous connections with special symplectic holonomy we classify all homogeneous symplectic manifolds with a torsion free connection of special symplectic holonomy i e a connection whose holonomy is an absolutely irreducible proper subgroup of the full symplectic group thereby we obtain many new explicit descriptions of manifolds with special symplectic holonomies we also show that manifolds with such a connection are homogeneous iff they contain no symmetric points and their symplectic scalar curvature is constant

operational rules and arbitrary order hermite generating functions operational identities often exploited in the theory of exponential operators are used to derive generating functions of hermite and kampe de feriet polynomials the method we propose allows us to obtain unilateral bilinear and bilateral generating functions by using the same procedure possible extensions of the technique are also discussed

structure from motion of rigid and jointed objects a method for recovering the three dimensional structure of moving rigid and jointed objects from several single camera views is presented the method is based on the fixed axis assumption all movement consists of translations and rotations about an axis that is fixed in direction for short periods of time this assumption makes it possible to recover the structure of any group of two or more rigidly connected points the structure of jointed objects is recovered by analyzing them as collections of rigid parts and then unifying the structures proposed for the parts the method presented here has been tested on several sets of data including movies used to demonstrate human perception of structure from motion

laguerre 2d functions and their application in quantum optics a set of orthonormalized and complete functions of two real variables in complex representation involving laguerre polynomials as a substantial part is introduced and is referred to as the set of laguerre two dimensional 2d functions the properties of the set of laguerre 2d functions are discussed and the fourier and radon transforms of these functions are calculated the laguerre 2d functions form a basis for a realization of the five dimensional lie algebra to the heisenberg weyl group for a two mode system real representation of this set of functions by a sum over products of hermite functions involving jacobi polynomials at the zero argument as coefficients is derived and it leads to new connections between laguerre and hermite polynomials in both directions the set of laguerre 2d functions is the most appropriate set of functions for the fock state representation of quasi probabilities in quantum optics the wigner quasi probability in fock state representation is up to a factor and an argument scaling directly given for each matrix element by a corresponding laguerre 2d function the properties of orthonormality and completeness of the laguerre 2d functions provide the fock state matrix elements of the density operator directly from the quasi probabilities the perina mista representation of the glauber sudarshan quasi probability can be represented with advantage by the laguerre 2d functions the fock state matrix elements of the displacement operator and the scalar product of displaced fock states are closely related to laguerre 2d functions

asymptotic formulae for two variable hermite polynomials new asymptotic formulae for the two dimensional hermite polynomials with large values of indices are found the applications to the photon distribution functions in squeezed one mode mixed quantum states are considered

image description and 3 d reconstruction from image trajectories of rotational motion a new technique for reconstructing the 3 d structure and motion of a scene undergoing relative rotational motion with respect to the camera is discussed given image correspondences of point features tracked over many frames a two stage technique for reconstruction is presented a grouping algorithm that exploits spatio temporal constraints of the common motion to achieve a reliable description of discrete point correspondences as curved trajectories in the image plane is developed in contrast trajectories fitted to points independent of each other lead to arbitrary image descriptions and very inaccurate 3 d parameters a new closed form solution under perspective projection for the 3 d motion and location of points from the computed image trajectories is also presented both stages are applied to real image sequences with good results this approach represents a first step in a longer term research effort examining the role of explicit spatio temporal organization in the interpretation of scenes from dynamic images

fourth order differential equation satisfied by the associated of any order of all classical orthogonal polynomials a study of their distribution of zeros abstract the first associated numerator polynomials of all classical orthogonal polynomials satisfy one fourth order differential equation valid for the four classical families but for the associated of arbitrary order the differential equations are only known separately in this work we introduce a program built in mathematica symbolic language which is able to construct the unique differential equation satisfied by the associated of any order of the classical class then we use this differential equation in order to study the distribution of zeros of these polynomials via their newton sum rules i e the sums of the kth power of zeros which are closely related with the moments of such a distribution

on a new family of hermite polynomials associated to parabolic cylinder functions we use the integral representation method to establish the link between hermite polynomials of ordinary nature and a family of recently introduced hermite polynomials associated with the parabolic cylinder functions

on the existence of infinite series of exotic holonomies in 1955 berger ber gave a list of irreducible reductive representations which can occur as the holonomy of a torsion free affine connection while this list was stated to be complete in the case of metric connections the situation in the general case remained unclear the representations which are missing from this list are called exotic in this paper we use twistor techniques to detect an infinite series of candidates for exotic holonomies we then develop a general method for constructing torsion free affine connections with prescribed holonomy which is based onequivariant deformations of a certain class of linear poisson structures when applied to the new series this method yields an exhaustive description of all torsion free connections with these holonomies and hence not only proves the existence of such connections but also allows us to deduce some striking facts about their local and global behaviour

fourth order differential equations satisfied by the generalized co recursive of all classical orthogonal polynomials a study of their distribution of zeros the unique fourth order differential equation satisfied by the generalized co recursive of all classical orthogonal polynomials is given for any but fixed level of recursivity up to now these differential equations were known only for each classical family separately and also for a specific recursivity level moreover we use this unique fourth order differential equation in order to study the distribution of zeros of these polynomials via their newton sum rules i e the sums of powers of their zeros which are closely related with the moments of such distribution both results are obtained with the help of two programs built in mathematica symbolic language

general hermite and laguerre two dimensional polynomials general hermite and laguerre two dimensional 2d polynomials which form a complex three parameter unification of the special hermite and laguerre 2d polynomials are defined and investigated the general hermite 2d polynomials are related to the two variable hermite polynomials but are not the same the advantage of the newly introduced hermite and laguerre 2d polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the two variable hermite polynomials two different sets of such polynomials are introduced which are biorthonormal to each other the matrix which plays a role in the new definition of hermite and laguerre 2d polynomials is in a considered sense the square root of the matrix which plays a role in the definition of two variable hermite polynomials two essentially different explicit representations of the hermite and laguerre 2d polynomials are derived where the first involves jacobi polynomials as coefficients in superpositions of special hermite or laguerre 2d polynomials and the second is a superposition of products of two hermite polynomials with decreasing indices and with coefficients related to the special laguerre 2d polynomials generating functions are derived for the hermite and laguerre 2d polynomials

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

transformations of laguerre 2d polynomials with applications to quasiprobabilities laguerre 2d polynomials are defined and their properties are investigated the laguerre 2d functions introduced in 1 2 are related to the laguerre 2d polynomials in such a way that they also include the weight function for the orthonormalization of the laguerre 2d polynomials a one parameter group of transformations applicable to certain classes of polynomials and discrete sets of functions is investigated and applied in particular to hermite polynomials and to laguerre 2d polynomials these transformations allow us to represent the polynomials of the corresponding classes by superpositions of the same kind of polynomials with stretched arguments they contain limiting cases with delta functions and their derivatives and lead to regularized representations of the delta functions and their derivatives as demonstrated for hermite and laguerre 2d polynomials applications of the laguerre 2d polynomials and 2d functions and their transformations to problems of quantum optics as the representation of quasiprobabilities in the fock state basis and by normally and otherwise ordered moments are considered the inversion of these representations is obtained in all cases a restricted design of quasiprobabilities should become possible

co recursive orthogonal polynomials and fourth order differential equations abstract we give explicitly the 4th order differential equation satisfied by the co recursive polynomials of the classical orthogonal polynomials we emphasize the legendre and bessel cases for which the co recursive differential equation coincide with the equation satisfied by the numerator polynomial differential equation and we propose new conjectures

operational identities and properties of ordinary and generalized special functions the theory of hermite laguerre and of the associated generating functions is reformulated within the framework of an operational formalism this point of view provides more efficient tools which allow the straightforward derivation of a wealth of new and old identities in this paper a central role is played by negative derivative operators and by their link with the tricomi functions and the generalized laguerre polynomials

on some generating functions for the extended generalized hermite polynomials in a recent paper g c greubel on some extensions and generating functions for the generalized hermite polynomials associated with functions of parabolic cylinder to be submitted the generalized hermite polynomials of wunsche were extended in this work we derive several generating functions for the extended hermite polynomials in terms of generalized lauricella functions in part i we derive several generating functions in part ii we derive an explicit relation for the extended generalized hermite polynomials in terms of the lauricella function ella function

connections with irreducible holonomy representations 3 irreducible berger algebras 17 3 1 real berger algebras 17 3 2 examples of berger algebras 18 3 2 1 conformal lie algebras 18 3 2 2 symmetric connections 20 3 2 3 symplectic lie algebras 21 3 2 4 complex lie algebras with h 6 0 23 3 3 irreducible complex berger algebras 25 3 4 simple complex berger algebras 29 3 5 complex tensor representations 39