polynomial approximation of certain biharmonic mappings abstract in this paper we first determine conditions on the analytic polynomials p and q such that the family of univalent sense preserving biharmonic polynomials of the form p z 2 p q is dense in the family of univalent sense preserving biharmonic mappings f defined on the unit disk d of the form f z 2 g next we consider the univalency of harmonic function g whenever f z 2 g is univalent in d finally we give a partial answer to the problem raised by muhanna about the radius of univalence of a family of sense preserving biharmonic mappings with the form f z 2 g h where g is univalent in d

numerical approximation of pde system fixed point map via newton s method abstract since the fundamental paper of moser 1966 it has been understood analytically that regularization is necessary as a postconditioning step in the application of approximate newton methods based upon the system differential map a development of these ideas in terms of current numerical methods and complexity estimates was given by the author 1985 it was proposed by the author 1989 to use the fixed point map as a basis for the linearization and thereby avoid the numerical loss of derivatives phenomenon identified by moser independently a coherent theory for the approximation of fixed points by numerical fixed points was devised by krasnosel skii and his coworkers 1972 in this paper the krasnosel skii calculus is merged with newton s method for the computation of the approximate fixed points in such a way that the approximation order is preserved with mesh independent constants since the application is to a system of partial differential equations the issue of the implicit nature of the linearized approximation must be addressed as well

an adaptive newton algorithm based on numerical inversion regularization as postconditioner we examine a class of approximate inversion processes satisfying estimates similar to those defined by finite element or truncated spectral approximations these are to be used as approximate right inverses for newton iteration methods when viewed at the operator level these approximations introduce a defect or loss of derivatives of order one or more regularization is introduced as a form of defect correction a superlinearly convergent approximate newton iteration is thereby obtained by using the numerical inversion adaptively i e with spectral or grid parameters correlated to the magnitude of the current residual in an intermediate norm defined by the defect this adaptive choice makes possible ascribing an order to the convergent process and this is identified as essentially optimal for elliptic problems relative to complexity the design of the algorithm involves multi parameter selection thereby opening up interesting avenues for elliptic problems relative to complexity this applies also to the regularization which may be carried out in the fourier transform space and is band limited in the language of whittaker shannon sampling theory the norms employed in the analysis are of holder space type the iteration is an adaptation of nash moser interation and the complexity studies use vituskin s theory of information processing computational experience is described in the final section

on certain geometric properties of polyharmonic mappings abstract the main purpose of this paper is twofold first the three quarter theorem is established for a large class of polyharmonic mappings second a class of polyharmonic mappings with positive jacobian and being univalent in the unit disk d are constructed

on the univalence of the log biharmonic mappings in this paper we establish the univalence and starlikeness connection between log biharmonic mappings and logharmonic mappings

on some properties of solutions of the biharmonic equation in the present paper the properties of the linear complex operator l f zfz z fz which is defined on the class of complex valued c1 functions in the plane are investigated it is shown that harmonicity and biharmonicity are invariant under the linear operatorl results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered the operator l can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way

bloch constant and landau s theorem for planar p harmonic mappings abstract in this paper our main aim is to introduce the concept of planar p harmonic mappings and investigate the properties of these mappings first we discuss the p harmonic bloch mappings two estimates on the bloch constant are obtained which are generalizations of the main results in colonna 1989 9 as a consequence of these investigations we establish a bloch and landau s theorem for p harmonic mappings

stokes flow in curved channels in this paper the problem of stokes flow in curved channels is solved by applying the matched eigenfunction expansion method the flow region is partitioned into a union of sectorial subregions then within each subregion the stream function is represented by an expansion of papkovich fadle eigenfunctions the coefficients in these expansions are obtained by requiring weak c 3 continuity of the stream function across common interfaces biorthogonality conditions in cylindrical coordinates are utilized

landau s theorem for log p harmonic mappings abstract the main aim of this paper is to prove the existence of landau bloch constant for log p harmonic mappings

landau s theorem for certain biharmonic mappings abstract in this paper we show the existence of landau and bloch constants for biharmonic mappings of the form l f here l represents the linear complex operator l z z z z defined on the class of complex valued c 1 functions in the plane and f belongs to the class of biharmonic mappings of the form f z z 2 g z k z z 1 where g and k are harmonic

biorthogonality condition for creeping flow in wedge shaped trenches in the present paper we derive the biorthogonality condition for stokes flow in wedge shaped trenches bounded by radial lines and concentric circles centered at the vertex of the wedge this condition is a property satisfied by the eigenfunctions and adjoint eigenfunctions needed to compute the coefficients of the eigenfunction expansion solution of the corresponding stokes flow problem

holomorphic jacobi manifolds and holomorphic contact groupoids this paper belongs to a series of works aiming at exploring generalized complex geometry in odd dimensions holomorphic jacobi manifolds were introduced and studied by the authors in a separate paper as special cases of generalized contact bundles in fact generalized contact bundles are nothing but odd dimensional analogues of generalized complex manifolds in the present paper we solve the integration problem for holomorphic jacobi manifolds by proving that they integrate to holomorphic contact groupoids a crucial tool in our proof is what we call the homogenization scheme which allows us to identify holomorphic jacobi manifolds with homogeneous holomorphic poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids

contact reduction and groupoid actions we introduce a new method to perform reduction of contact manifolds that extends willett s and albert s results to carry out our reduction procedure all we need is a complete jacobi map j m 0 from a contact manifold to a jacobi manifold this naturally generates the action of the contact groupoid of 0 on m and we show that the quotients of fibers j 1 x by suitable lie subgroups x are either contact or locally conformal symplectic manifolds with structures induced by the one on m we show that willett s reduced spaces are prequantizations of our reduced spaces hence the former are completely determined by the latter since a symplectic manifold is prequantizable iff the symplectic form is integral this explains why willett s reduction can be performed only at distinguished points as an application we obtain kostant s prequantizations of coadjoint orbits finally we present several examples where we obtain classical contact manifolds as reduced spaces

integration of twisted dirac brackets given a lie groupoid g over a manifold m we show that multiplicative 2 forms on g relatively closed with respect to a closed 3 form phi on m correspond to maps from the lie algebroid of g into t m satisfying an algebraic condition and a differential condition with respect to the phi twisted courant bracket this correspondence describes as a special case the global objects associated to phi twisted dirac structures as applications we relate our results to equivariant cohomology and foliation theory and we give a new description of quasi hamiltonian spaces and group valued momentum maps

landau s theorem for polyharmonic mappings abstract in this paper we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk d then we obtain two versions of landau s theorem for polyharmonic mappings f and for the mappings of the type l f where l is the differential operator of abdulhadi abu muhanna and khuri examples and numerical estimates are given

biorthogonality condition for axisymmetric stokes flow in a toroidal region in the present paper the biorthogonality condition for axisymmetric stokes flow in a toroidal geometry is derived this condition is needed to compute the coefficients of the eigenfunction expansion solution of the corresponding stokes flow problem

newton s method for gradient equations based upon the fixed point map convergence and complexity study an approximate newton method based upon the fixed point mapt is introduced for scalar gradient equations although the exact newton method coincides for such scalar equations with the standard iteration the structure of the fixed point map provides a way of defining anr quadratically convergent finite element iteration in the spirit of the kantorovich theory the loss of derivatives phenomenon typically experienced in approximate newton methods is thereby avoided it is found that two grid parameters are sssential h and bar h approx h 2 the latter is used to calculate the approximate residual and is isolated as a fractional step it is equivalent to the approximation oft the former is used to calculate the newton increment and this is equivalent to the approximation oft the complexity of the finite element computation for the newton increment is shown to be of optimal order via the vituskin inequality relating metric entropy andn widths

schwarzian derivative integral means and the affine and linear invariant families of biharmonic mappings abstract in this paper we discuss the properties of the schwarzian derivative integral means and the affine and linear invariant families of biharmonic mappings first we introduce the schwarzian derivative s f for biharmonic mappings f z 2g h and obtain several necessary and sufficient conditions for s f to be analytic second we introduce the subordination of biharmonic mappings and obtain inequalities for integral means of subordinate biharmonic mappings finally we introduce the affine and linear invariant families of biharmonic mappings and prove several estimates related to the jacobian of functions in these invariant families

convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh canonical edge problems for the biharmonic equation can be solved by separating variables the eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of r c smith 1952 we study the reduced system which is governed by a vector ordinary differential equation a solution of the biharmonic problem governed by a partial differential equation can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem the theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f y with values in 2 can be represented in terms of eigenvectors of the reduced problem this paper adds new theorems and conjectures to the theory we extend smith s generalization to fourth order problems of the methods introduced by titchmarsh 1946 to study eigenfunction expansions associated with second order problems we use this method to prove that if f y f1 y f2y 1 y 1 f y e c1 1 1 f e l2 1 1 then the series expressing f y converges uniformly to f y in the open interval 1 1 uniformly in 1 1 if f1 1 0 and in any case to 0 f2 1 f1 1 at y 1 this is unlike fourier series which converge to the mean value of the periodic extension of a function the series exhibits a gibbs phenomenon near the end points of discontinuity when f1 1 0

a note on the eigenfunction expansion for the elastic strip the papkovitch fadle expansion of the biharmonic stress function for a semi infinite strip x leqq 1 y geqq 0 with free edges x pm 1 is obtained as the residue sum of a fourier integral for the case of end data corresponding to prescribed normal stress and transverse displacement which implies knowledge of phi xx x 0 equiv f 2 x and delta phi x 0 equiv f 4 x it is shown that if f f 2 f 4 is twice differentiable with bf f in l 2 0 1 and the data are compatible with the edge conditions at the corners x pm 1 y 0 the derived expansions for phi xx x y and delta phi x y converge to f 2 x f 4 x as y downarrow 0 for x on the closed interval 0 1 even when the compatibility conditions are not met the expansions converge to the data on the open interval 0 1

morita equivalence of poisson manifolds poisson manifolds are the classical analogue of associative algebras for poisson manifolds symplectic realizations play a similar role as representations do for associative algebras in this paper the notion of morita equivalence of poisson manifolds a classical analogue of morita equivalence of algebras is introduced and studied it is proved that morita equivalent poisson manifolds have equivalent categories of complete symplectic realizations for certain types of poisson manifolds the geometric invariants of morita equivalence are also investigated

biorthogonal series solution of stokes flow problems in sectorial regions in this paper we develop an eigenfunction expansion method for solving stokes flow problems in sectorial cavities that arise in fluid dynamics the method leads to the development of a set of eigenfunctions adjoint eigenfunctions biorthogonality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansion the resulting infinite system of linear equations is then solved by truncation these biorthogonality conditions are properties satisfied by the eigenfunctions and adjoint eigenfunctions which are used to compute the coefficients of the eigenfunction expansion solution the biorthogonality conditions are derived for a class of fourth order boundary value problems with variable coefficients that arise from separating variables of the governing stokes equation the method is applied to the slow steady flow in a two dimensional sectorial cavity the fluid is set into motion by the uniform translation of a covering plate or belt