discrete subgroups of mobius transformations abstract in this paper we present the necessary geometric conditions for a nonelementary two generator subgroup of mobius transformations acting in r n r n to be discrete

computational properties of three term recurrence relations for kummer functions several three term recurrence relations for confluent hypergeometric functions are analyzed from a numerical point of view minimal and dominant solutions for complex values of the variable z are given derived from asymptotic estimates of the whittaker functions with large parameters the laguerre polynomials and the regular coulomb wave functions are studied as particular cases with numerical examples of their computation

discreteness of subgroups of pu 2 1 with regular elliptic elements abstract in y p jiang s kamiya j r parker jogensen s inequality for complex hyperbolic space geometriae dedicate 97 2003 55 80 jiang et al provided jorgensen s inequality for non elementary group of isometries of complex hyperbolic space generated by two elements one of which is loxodromic or boundary elliptic in this paper we give analogues of jorgensen s inequality for the subgroup generated by two elements but one of which is regular elliptic

a new parallel solver suited for arbitrary semilinear parabolic partial differential equations based on generalized random trees a probabilistic representation for initial value semilinear parabolic problems based on generalized random trees has been derived two different strategies have been proposed both requiring generating suitable random trees combined with a pade approximant for approximating accurately a given divergent series such series are obtained by summing the partial contribution to the solution coming from trees with arbitrary number of branches the new representation greatly expands the class of problems amenable to be solved probabilistically and was used successfully to develop a generalized probabilistic domain decomposition method such a method has been shown to be suited for massively parallel computers enjoying full scalability and fault tolerance finally a few numerical examples are given to illustrate the remarkable performance of the algorithm comparing the results with those obtained with a classical method

a stochastic approach to the solution of magnetohydrodynamic equations the construction of stochastic solutions is a powerful method to obtain localized solutions in configuration or fourier space and for parallel computation with domain decomposition here a stochastic solution is obtained for the magnetohydrodynamics equations some details are given concerning the numerical implementation of the solution which is illustrated by an example of generation of long range magnetic fields by a velocity source

stochastic cascades and 3 dimensional navier stokes equations in this article we study the incompressible navier stokes equations in 3 the non linear integral equation satisfied by the fourier transform of the laplacian of the velocity field can be interpreted in terms of a branching process and a composition rule along the associated tree we derive from this representation new classes where global existence and uniqueness can be proven

friedland hersonsky problem for matrix algebra abstract in this paper we show that the answer to problem 3 9 in duke math j 69 1993 593 is positive when n 2 and negative when n 3

a discreteness criterion for subgroups of pu 2 1 with screw parabolic elements abstract we establish a discreteness criterion for pu 2 1 with a screw parabolic element g which can be viewed as a generalization of shimizu s lemma our method relies upon the geometry of margulis region it allows us to get rid of the restriction on rotation angle 2 of g in 4 7 besides an explicit region precisely invariant under the stabilizer of the fixed point of g in is given

discrete groups of unitary isometries and balls in hyperbolic manifolds abstract we study discrete groups of unitary isometries in c algebras and revisit the bieberbach groups we give lower estimates of radii of balls in complete hyperbolic manifolds with finite volumes

algebraic convergence of finitely generated kleinian groups in all dimensions abstract in this paper we prove that for a sequence g i r of r generator kleinian groups acting on r n if g i r satisfies condition a then its algebraic limit is also a kleinian group this generalizes the main result in x wang algebraic convergence theorems of n dimensional kleinian groups israel j math 162 2007 221 233

highly efficient numerical algorithm based on random trees for accelerating parallel vlasov poisson simulations an efficient numerical method based on a probabilistic representation for the vlasov poisson system of equations in the fourier space has been derived this has been done theoretically for arbitrary dimensional problems and particularized to unidimensional problems for numerical purposes such a representation has been validated theoretically in the linear regime comparing the solution obtained with the classical results of the linear landau damping the numerical strategy followed requires generating suitable random trees combined with a pade approximant for approximating accurately a given divergent series such series are obtained by summing the partial contributions to the solution coming from trees with arbitrary number of branches these contributions coming in general from multi dimensional definite integrals are efficiently computed by a quasi monte carlo method it is shown how the accuracy of the method can be effectively increased by considering more terms of the series the new representation was used successfully to develop a probabilistic domain decomposition method suited for massively parallel computers which improves the scalability found in classical methods finally a few numerical examples based on classical phenomena such as the non linear landau damping and the two streaming instability are given illustrating the remarkable performance of the algorithm when compared the results with those obtained using a classical method

smooth or singular solutions to the navier stokes system abstract the existence of singular solutions of the incompressible navier stokes system with singular external forces the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions including stationary ones and a pointwise loss of smoothness for solutions are proved in the same function space of pseudomeasure type

domain decomposition solution of nonlinear two dimensional parabolic problems by random trees a domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension based on the probabilistic representation of solutions as an average of suitable multiplicative functionals such a direct probabilistic representation requires generating a number of random trees whose role is that of the realizations of stochastic processes used in the linear problems first only few values of the sought solution inside the space time domain are computed by a monte carlo method on the trees an interpolation is then carried out in order to approximate interfacial values of the solution inside the domain thus a fully decoupled set of sub problems is obtained the algorithm is suited to massively parallel implementation enjoying arbitrary scalability and fault tolerance properties pruning the trees is shown to increase appreciably the efficiency of the algorithm numerical examples conducted in 2d including some for the kpp equation are given

orientation preserving mobius transformations in r 4 and quaternionic determinants abstract this paper explores m r 4 the group of orientation preserving mobius transformations acting in r 4 on the one hand m r 4 is given by the group of 2 2 matrices over the quaternions h with determinant d derived from the corresponding 4 4 matrices over the complex numbers c on the other hand we know that in general m r n may be given in terms of 2 2 matrices over the clifford algebra c n with n 1 generators thus when n 4 m r 4 is given in terms of 2 2 matrices with entries drawn from c 4 and determinant defined in terms of the entries of the given matrix we note that the skew field h may be considered as a clifford algebra c 3 based on two generators i and j or more generally i 1 and i 2 where i j k or i 1 i 2 k while the set of elements 1 i j k form a basis of h regarded as a 4 dimensional real vector space thus h is embedded in c 4 in the paper we reconcile the two representations of m r 4 by comparing the generating sets of the underlying groups of matrices a relationship between a determinants d and is also exposed

groups generated by two elliptic elements in pu 2 1 abstract et f and g be two elliptic elements in pu 2 1 we prove that if the distance f g between the complex lines or points fixed by f and g is large than a certain number then the group f g is discrete non elementary and isomorphic to the free product f g

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

coulomb and bessel functions of complex arguments and order abstract the coulomb wavefunctions originally constructed for real 0 real and integer 0 are defined for and all complex we examine the complex continuation of a variety of series and continued fraction expansions for the coulomb functions and their logarithmic derivatives and then see how these expansions may be selectively combined to calculate both the regular and irregular functions and their derivatives the resulting algorithm 46 is a complex generalisation of steed s method 6 7 as it appears in the real procedure coulfg 10 complex whittaker confluent hypergeometric and bessel functions can also be calculated

a rate of convergence for the lansα regularization of navier stokes equations abstract a rate of convergence of the solutions of the lans equations with periodic boundary to the solutions of the navier stokes equations as 0 is obtained in a mixed l 1 l 2 time space norm for small initial data in besov type function spaces in which global existence and uniqueness of solutions can also be established

mobius gyrogroups a clifford algebra approach abstract using the clifford algebra formalism we study the mobius gyrogroup of the ball of radius t of the paravector space r v where v is a finite dimensional real vector space we characterize all the gyro subgroups of the mobius gyrogroup and we construct left and right factorizations with respect to an arbitrary gyro subgroup for the paravector ball the geometric and algebraic properties of the equivalence classes are investigated we show that the equivalence classes locate in a k dimensional sphere where k is the dimension of the gyro subgroup and the resulting quotient spaces are again mobius gyrogroups with the algebraic structure of the factorizations we study the sections of mobius fiber bundles inherited by the mobius projectors

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

delambre gauss formulas for augmented right angled hexagons in hyperbolic 4 space abstract we study the geometry of right angled hexagons in the hyperbolic 4 space h 4 via clifford numbers or quaternions we show how to augment alternate sides of such a hexagon and arbitrarily orient each line and plane involved so that for the non augmented sides we can define quaternion half side lengths whose angular parts are obtained from half the euler angles associated to a certain orientation preserving isometry of the euclidean 3 space we also define appropriate complex half side lengths for the augmented sides of the augmented hexagon we further explain how to geometrically read off the quaternion half side lengths for a given oriented augmented right angled hexagon in h 4 our main result is a set of generalized delambre gauss formulas for oriented augmented right angled hexagons in h 4 involving the quaternion half side lengths and the complex half side lengths we also show in the appendix how the same method gives delambre gauss formulas for oriented right angled hexagons in h 3 from which the well known laws of sines and of cosines can be deduced these formulas generalize the classical delambre gauss formulas for spherical hyperbolic triangles

quasiconformal homogeneity of hyperbolic manifolds we exhibit strong constraints on the geometry and topology of a uniformly quasiconformally homogeneous hyperbolic manifold in particular if n 3 a hyperbolic n manifold is uniformly quasiconformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold moreover if n 3 we show that there is a constant kn 1 such that if m is a hyperbolic n manifold other than open image in new window which is k quasiconformally homogeneous then k kn

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

majorizing kernels and stochastic cascades with applications to incompressible navier stokes equations a general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d navier stokes equations the basic idea generalizes a probabilistic approach introduced by lejan and sznitman 1997 to obtain weak solutions whose fourier transform may be represented by an expected value of a stochastic cascade a functional analytic framework is also developed which partially connects stochastic iterations and certain picard iterates some local existence and uniqueness results are also obtained by contractive mapping conditions on the picard iteration

on the uniform equidistribution of closed horospheres in hyperbolic manifolds we prove equidistribution results for pieces of closed horospheres in cofinite hyperbolic manifolds of dimension n 1 using spectral methods this extends earlier results by hejhal hej2 and str

uniform discreteness and heisenberg screw motions we give aversion of shimizu s lemma for groups of complex hyperbolic isometries one of whose generators is aheisenberg screw motion our main result interpolates between known results for groups with a generator that is avertical translation or aheisenberg rotation we also give an interpretation of our result in terms of the relation between radii of isometric spheres and their distance from the axis of the heisenberg screw motion

transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions three term recurrence relations yn 1 bnyn anyn 1 0 can be used for computing recursively a great number of special functions depending on the asymptotic nature of the function to be computed different recursion directions need to be considered backward for minimal solutions and forward for dominant solutions however some solutions interchange their role for finite values of n with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal this phenomenon related to gautschi s anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions is shown to be a general situation which takes place for recurrences with an negative and bn changing sign once we analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations modified bessel functions confluent and gauss hypergeometric functions and discuss the implication of such transitory behaviour on the numerical stability of recursion

identifying minimal and dominant solutions for kummer recursions we identify minimal and dominant solutions of three term recurrence relations for the confluent hypergeometric functions 1f1 a 1n c 2n z and u a 1n c 2n z where i 0 1 not both equal to 0 the results are obtained by applying perron s theorem together with uniform asymptotic estimates derived by t m dunster for whittaker functions with large parameter values the approximations are valid for complex values of a c and z with ph z

multiplicative cascades applied to pdes two numerical examples numerical approximations to the fourier transformed solution of partial differential equations are obtained via monte carlo simulation of certain random multiplicative cascades two particular equations are considered linear diffusion equation and viscous burgers equation the algorithms proposed exploit the structure of the branching random walks in which the multiplicative cascades are defined the results show initial numerical approximations with errors less than 5 in the leading fourier coefficients of the solution this approximation is then improved substantially using a picard iteration scheme on the integral equation associated with the representation of the respective pde in fourier space

a probabilistic representation of solutions of the incompressible navier stokes equations in r3 a new probabilistic representation is presented for solutions of the incompressible navier stokes equations in r3 with given forcing and initial velocity this representation expresses solutions as scaled conditional expectations of functionals of a markov process indexed by the nodes of a binary tree it gives existence and uniqueness of weak solutions for all time under relatively simple conditions on the forcing and initial data these conditions involve comparison of the forcing and initial data with majorizing kernels